Data Pre-processing

Load needed libraries

library(fastDummies)
library(readr)
library(ggplot2)
library(dplyr)
library(caret)
library(glmnet)
library(boot)
library(tree)
library(ranger)
library(xgboost)
library(gbm)
library(vip)
library(ISLR)
library(tidyr)
library(gridExtra)
library(reshape2)

Set the seed for reproducibility

set.seed(1)

Load the dataset

original_lc_data <- read.csv("LCdata.csv",sep = ";")
lc_data <- original_lc_data

Remove attributes not available for prediction

lc_data <- subset(lc_data, select = -c(collection_recovery_fee, installment, issue_d,
                                       last_pymnt_amnt, last_pymnt_d, loan_status,
                                       next_pymnt_d, out_prncp, out_prncp_inv,
                                       pymnt_plan, recoveries, total_pymnt,
                                       total_pymnt_inv,total_rec_int, total_rec_late_fee,                                                  total_rec_prncp))
summary(lc_data)
       id             member_id          loan_amnt      funded_amnt    funded_amnt_inv     term              int_rate    
 Min.   :   54734   Min.   :   70473   Min.   :  500   Min.   :  500   Min.   :    0   Length:798641      Min.   : 5.32  
 1st Qu.: 9207230   1st Qu.:10877939   1st Qu.: 8000   1st Qu.: 8000   1st Qu.: 8000   Class :character   1st Qu.: 9.99  
 Median :34433372   Median :37095300   Median :13000   Median :13000   Median :13000   Mode  :character   Median :12.99  
 Mean   :32463636   Mean   :35000265   Mean   :14754   Mean   :14741   Mean   :14702                      Mean   :13.24  
 3rd Qu.:54900100   3rd Qu.:58470266   3rd Qu.:20000   3rd Qu.:20000   3rd Qu.:20000                      3rd Qu.:16.20  
 Max.   :68617057   Max.   :73544841   Max.   :35000   Max.   :35000   Max.   :35000                      Max.   :28.99  
                                                                                                                         
  emp_title          emp_length        home_ownership       annual_inc      verification_status     url           
 Length:798641      Length:798641      Length:798641      Min.   :      0   Length:798641       Length:798641     
 Class :character   Class :character   Class :character   1st Qu.:  45000   Class :character    Class :character  
 Mode  :character   Mode  :character   Mode  :character   Median :  65000   Mode  :character    Mode  :character  
                                                          Mean   :  75014                                         
                                                          3rd Qu.:  90000                                         
                                                          Max.   :9500000                                         
                                                          NA's   :4                                               
     desc             purpose             title             zip_code          addr_state             dti         
 Length:798641      Length:798641      Length:798641      Length:798641      Length:798641      Min.   :   0.00  
 Class :character   Class :character   Class :character   Class :character   Class :character   1st Qu.:  11.91  
 Mode  :character   Mode  :character   Mode  :character   Mode  :character   Mode  :character   Median :  17.66  
                                                                                                Mean   :  18.16  
                                                                                                3rd Qu.:  23.95  
                                                                                                Max.   :9999.00  
                                                                                                                 
  delinq_2yrs      earliest_cr_line   inq_last_6mths    mths_since_last_delinq mths_since_last_record    open_acc    
 Min.   : 0.0000   Length:798641      Min.   : 0.0000   Min.   :  0.0          Min.   :  0.0          Min.   : 0.00  
 1st Qu.: 0.0000   Class :character   1st Qu.: 0.0000   1st Qu.: 15.0          1st Qu.: 51.0          1st Qu.: 8.00  
 Median : 0.0000   Mode  :character   Median : 0.0000   Median : 31.0          Median : 70.0          Median :11.00  
 Mean   : 0.3145                      Mean   : 0.6947   Mean   : 34.1          Mean   : 70.1          Mean   :11.55  
 3rd Qu.: 0.0000                      3rd Qu.: 1.0000   3rd Qu.: 50.0          3rd Qu.: 92.0          3rd Qu.:14.00  
 Max.   :39.0000                      Max.   :33.0000   Max.   :188.0          Max.   :129.0          Max.   :90.00  
 NA's   :25                           NA's   :25        NA's   :408818         NA's   :675190         NA's   :25     
    pub_rec          revol_bal         revol_util       total_acc      initial_list_status last_credit_pull_d
 Min.   : 0.0000   Min.   :      0   Min.   :  0.00   Min.   :  1.00   Length:798641       Length:798641     
 1st Qu.: 0.0000   1st Qu.:   6443   1st Qu.: 37.70   1st Qu.: 17.00   Class :character    Class :character  
 Median : 0.0000   Median :  11876   Median : 56.00   Median : 24.00   Mode  :character    Mode  :character  
 Mean   : 0.1953   Mean   :  16930   Mean   : 55.05   Mean   : 25.27                                         
 3rd Qu.: 0.0000   3rd Qu.:  20839   3rd Qu.: 73.50   3rd Qu.: 32.00                                         
 Max.   :63.0000   Max.   :2904836   Max.   :892.30   Max.   :169.00                                         
 NA's   :25        NA's   :2         NA's   :454      NA's   :25                                             
 collections_12_mths_ex_med mths_since_last_major_derog  policy_code application_type   annual_inc_joint   dti_joint     
 Min.   : 0.00000           Min.   :  0.0               Min.   :1    Length:798641      Min.   : 17950   Min.   : 3.0    
 1st Qu.: 0.00000           1st Qu.: 27.0               1st Qu.:1    Class :character   1st Qu.: 76167   1st Qu.:13.3    
 Median : 0.00000           Median : 44.0               Median :1    Mode  :character   Median :101886   Median :17.7    
 Mean   : 0.01447           Mean   : 44.1               Mean   :1                       Mean   :110745   Mean   :18.4    
 3rd Qu.: 0.00000           3rd Qu.: 61.0               3rd Qu.:1                       3rd Qu.:133000   3rd Qu.:22.6    
 Max.   :20.00000           Max.   :188.0               Max.   :1                       Max.   :500000   Max.   :43.9    
 NA's   :126                NA's   :599107                                              NA's   :798181   NA's   :798183  
 verification_status_joint acc_now_delinq       tot_coll_amt      tot_cur_bal       open_acc_6m       open_il_6m    
 Length:798641             Min.   : 0.000000   Min.   :      0   Min.   :      0   Min.   : 0.0     Min.   : 0.0    
 Class :character          1st Qu.: 0.000000   1st Qu.:      0   1st Qu.:  29861   1st Qu.: 0.0     1st Qu.: 1.0    
 Mode  :character          Median : 0.000000   Median :      0   Median :  80647   Median : 1.0     Median : 2.0    
                           Mean   : 0.005026   Mean   :    228   Mean   : 139508   Mean   : 1.1     Mean   : 2.9    
                           3rd Qu.: 0.000000   3rd Qu.:      0   3rd Qu.: 208229   3rd Qu.: 2.0     3rd Qu.: 4.0    
                           Max.   :14.000000   Max.   :9152545   Max.   :8000078   Max.   :14.0     Max.   :33.0    
                           NA's   :25          NA's   :63276     NA's   :63276     NA's   :779525   NA's   :779525  
  open_il_12m      open_il_24m     mths_since_rcnt_il  total_bal_il       il_util        open_rv_12m      open_rv_24m    
 Min.   : 0.0     Min.   : 0.0     Min.   :  0.0      Min.   :     0   Min.   :  0.0    Min.   : 0.0     Min.   : 0      
 1st Qu.: 0.0     1st Qu.: 0.0     1st Qu.:  6.0      1st Qu.: 10164   1st Qu.: 58.4    1st Qu.: 0.0     1st Qu.: 1      
 Median : 0.0     Median : 1.0     Median : 12.0      Median : 24544   Median : 74.8    Median : 1.0     Median : 2      
 Mean   : 0.8     Mean   : 1.7     Mean   : 21.1      Mean   : 36428   Mean   : 71.5    Mean   : 1.4     Mean   : 3      
 3rd Qu.: 1.0     3rd Qu.: 2.0     3rd Qu.: 23.0      3rd Qu.: 47640   3rd Qu.: 87.7    3rd Qu.: 2.0     3rd Qu.: 4      
 Max.   :12.0     Max.   :19.0     Max.   :363.0      Max.   :878459   Max.   :223.3    Max.   :22.0     Max.   :43      
 NA's   :779525   NA's   :779525   NA's   :780030     NA's   :779525   NA's   :782007   NA's   :779525   NA's   :779525  
   max_bal_bc        all_util      total_rev_hi_lim      inq_fi        total_cu_tl      inq_last_12m   
 Min.   :    0    Min.   :  0.0    Min.   :      0   Min.   : 0.0     Min.   : 0.0     Min.   :-4      
 1st Qu.: 2406    1st Qu.: 47.6    1st Qu.:  13900   1st Qu.: 0.0     1st Qu.: 0.0     1st Qu.: 0      
 Median : 4502    Median : 61.9    Median :  23700   Median : 0.0     Median : 0.0     Median : 2      
 Mean   : 5878    Mean   : 60.8    Mean   :  32093   Mean   : 0.9     Mean   : 1.5     Mean   : 2      
 3rd Qu.: 7774    3rd Qu.: 75.2    3rd Qu.:  39800   3rd Qu.: 1.0     3rd Qu.: 2.0     3rd Qu.: 3      
 Max.   :83047    Max.   :151.4    Max.   :9999999   Max.   :16.0     Max.   :35.0     Max.   :32      
 NA's   :779525   NA's   :779525   NA's   :63276     NA's   :779525   NA's   :779525   NA's   :779525  

First we delete the columns which aren’t useful for our prediction

lc_data$id <- NULL
lc_data$member_id <- NULL
lc_data$zip_code <- NULL
lc_data$url <- NULL

Looks like policy_code contains just value equal to 1, it can be removed

lc_data$policy_code <- NULL

Remove additional columns which are related to the historical data

lc_data$last_credit_pull_d <- NULL

Then we delete the columns which can’t be converted to categorical and require NLP

lc_data$title <- NULL
lc_data$desc <- NULL
lc_data$emp_title <- NULL

Let’s examine the loan_amnt column

sum(is.na(lc_data$loan_amnt))
[1] 0
cor(lc_data$loan_amnt, lc_data$int_rate)
[1] 0.1447189
hist(lc_data$loan_amnt, breaks = 20, main = "loan_amnt distribution", xlab = "loan_amnt", col = "lightblue", border = "black")

ggplot(data = lc_data, mapping = aes(x=int_rate,y=loan_amnt)) + geom_boxplot()

Standardize loan_amnt

#lc_data$loan_amnt <- scale(lc_data$loan_amnt)

Let’s examine the funded_amnt column

sum(is.na(lc_data$funded_amnt))
[1] 0
cor(lc_data$funded_amnt, lc_data$int_rate)
[1] 0.1448634
hist(lc_data$funded_amnt, breaks = 20, main = "funded_amnt distribution", xlab = "funded_amnt", col = "lightblue", border = "black")

As we can see, funded_amnt is almost the same as the loan_amnt column, consequently, we remove it.

lc_data$funded_amnt <- NULL 

Let’s examine the funded_amnt_inv column

sum(is.na(lc_data$funded_amnt_inv))
[1] 0
cor(lc_data$funded_amnt_inv, lc_data$int_rate)
[1] 0.1449083
hist(lc_data$funded_amnt_inv, breaks = 20, main = "funded_amnt_inv distribution", xlab = "funded_amnt_inv", col = "lightblue", border = "black")

Remove funded_amnt_inv for the same reason as above

lc_data$funded_amnt_inv <- NULL

Let’s see the int_rate distribution.

hist(lc_data$int_rate, breaks = 20, main = "int_rate distribution", xlab = "int_rate", col = "lightblue", border = "black")

Standardize int rate:

#lc_data$int_rate <- scale(lc_data$int_rate)

As we can observe, there are 40363 NAs. We can assume 40363 do not work.

barplot(table(lc_data$emp_length),
        xlab = "emp_length years", 
        ylab = "Frequency", 
        col = "skyblue", 
        border = "black",
        cex.names = 0.6)  # The size of the main title

Since emp_length seems to be categorical, we transform it to as a factor and then as numeric. The conversion to numeric is needed for supporting the XGBoost

lc_data$emp_length <- as.factor(lc_data$emp_length)
ggplot(data = lc_data, mapping = aes(x=int_rate,y=emp_length)) + geom_boxplot()

lc_data$emp_length <- as.numeric(lc_data$emp_length)

As we can see, term plays a crucial role in predicting the interest rate.

lc_data$term <- as.factor(lc_data$term)
ggplot(data = lc_data, mapping = aes(x=int_rate,y=term)) + geom_boxplot()

lc_data$term <- as.numeric(lc_data$term)

Cleaning of home_ownership:

During the data cleaning phase, our analysis revealed that the variable “home_ownership” does not show a distinct correlation with interest rates. Specifically, among the categories, “ANY” and “OTHER” contain 2 and 154 cases, respectively, while the “NONE” category comprises 39 cases. Although the “NONE” category appears to demonstrate a higher interest rate compared to others, the limited sample size of 39 cases raises doubts about the reliability of this observation. Notably, the “NONE” category might pertain to individuals experiencing homelessness, prompting ethical concerns about loan provision to this demographic.

table(lc_data$home_ownership)

     ANY MORTGAGE     NONE    OTHER      OWN     RENT 
       2   399151       45      155    78789   320499 
ggplot(data = lc_data, mapping = aes(x=int_rate,y=home_ownership)) + geom_boxplot()

Then, we retain mortgage, own and rent:

lc_data <- lc_data %>% filter(home_ownership %in% c("MORTGAGE","OWN","RENT"))
lc_data$home_ownership <- as.numeric(as.factor(lc_data$home_ownership))

Application joint handling:


# merging annual income
lc_data <- lc_data %>% mutate(
    annual_inc_merged = ifelse(is.na(annual_inc_joint)== TRUE, annual_inc,annual_inc_joint)) 

lc_data <- lc_data %>% select(-annual_inc,-annual_inc_joint)

# merging debt to income ratio
lc_data <- lc_data %>% mutate(
    dti_merged = ifelse(is.na(dti_joint)== TRUE, dti,dti_joint)) 

lc_data <- lc_data %>% select(-dti,-dti_joint)

Upon reviewing the summary again, it becomes apparent that there are merely 460 joint applications, constituting a small subset within the extensive dataset of around 800k rows. Through consolidating the debt-to-income ratios (dti’s), we can pinpoint the data pertinent to our research objectives. Hence, it is advisable to eliminate the columns verification_status_joint and application_type to prevent introducing unwarranted variability into our analysis.

table(lc_data$verification_status)

   Not Verified Source Verified        Verified 
         240255          296631          261553 
table(lc_data$verification_status_joint)

                   Not Verified Source Verified        Verified 
         797979             253              53             154 
lc_data$verification_status <- as.numeric(as.factor(lc_data$verification_status))
lc_data <- lc_data %>% select(-verification_status_joint, -application_type)

Let’s check if other is NA or a real value for purpose. It’s a real one, so we don’t have to handle it.

lc_data$purpose <- as.factor(lc_data$purpose)
ggplot(data = lc_data, mapping = aes(x=int_rate,y=purpose)) + geom_boxplot()

lc_data$purpose <- as.numeric(lc_data$purpose)

Let’s have a glance to the state address:

table(lc_data$addr_state)

    AK     AL     AR     AZ     CA     CO     CT     DC     DE     FL     GA     HI     IA     ID     IL     IN     KS 
  1992  10101   5953  18359 116578  16934  12154   2188   2268  54819  26146   4112     13     11  31880  12393   7105 
    KY     LA     MA     MD     ME     MI     MN     MO     MS     MT     NC     ND     NE     NH     NJ     NM     NV 
  7726   9498  18546  18906    469  20678  14306  12821   3455   2286  22135    431   1064   3865  29991   4428  11155 
    NY     OH     OK     OR     PA     RI     SC     SD     TN     TX     UT     VA     VT     WA     WI     WV     WY 
 66790  26682   7266   9806  28221   3499   9609   1615  11618  63982   5629  23616   1606  17470  10446   3977   1841 
lc_data$addr_state <- as.factor(lc_data$addr_state)
ggplot(data = lc_data, mapping = aes(x=int_rate,y=addr_state)) + geom_boxplot()

lc_data$addr_state <- as.numeric(lc_data$addr_state)

Regarding delinquency in the last 2 years, there are few NAs then remove them:

lc_data <- lc_data %>% 
    filter(!(is.na(delinq_2yrs)))

The columns mths_since_delinq_cat, mths_since_last_record, mths_since_rcnt_il and mths_since_last_major_derog contain numerical values which refer to the number of the months. Since this columns contain a lot of null values which can’t be replaced with 0’s, one of the most appropriate operations that can be made is applying discretization. We do this by creating a set of contiguous bins based on years, while for the null values we create a separate bin.

lc_data <- lc_data %>%
  mutate(mths_since_delinq_cat = ifelse(
    is.na(mths_since_last_delinq) == TRUE,
    "NONE",
    ifelse(
      mths_since_last_delinq <= 12,
      "Less_1_Y",
      ifelse(
        mths_since_last_delinq <= 24,
        "Less_2_Y",
        ifelse(
          mths_since_last_delinq <= 36,
          "Less_3_Y",
          ifelse(mths_since_last_delinq <= 48, "Less_4_Y", "More_4_Y")
        )
      )
    )
  )) %>% select(-mths_since_last_delinq)
          
lc_data$mths_since_delinq_cat <- as.factor(lc_data$mths_since_delinq_cat)
ggplot(data = lc_data, mapping = aes(x=int_rate,y=mths_since_delinq_cat))+geom_boxplot()

lc_data$mths_since_delinq_cat <- as.numeric(lc_data$mths_since_delinq_cat)
lc_data <- lc_data %>%
  mutate(mths_since_last_record_cat = ifelse(
    is.na(mths_since_last_record) == TRUE,
    "NONE",
    ifelse(
      mths_since_last_record <= 12,
      "Less_1_Y",
      ifelse(
        mths_since_last_record <= 24,
        "Less_2_Y",
        ifelse(
          mths_since_last_record <= 36,
          "Less_3_Y",
          ifelse(mths_since_last_record <= 48, "Less_4_Y", "More_4_Y")
        )
      )
    )
  )) %>% select(-mths_since_last_record)

lc_data$mths_since_last_record_cat <- as.factor(lc_data$mths_since_last_record_cat)
ggplot(data = lc_data, mapping = aes(x=int_rate,y=mths_since_last_record_cat))+geom_boxplot()

lc_data$mths_since_last_record_cat <- as.numeric(lc_data$mths_since_last_record_cat)
lc_data <-lc_data %>% 
  mutate(mths_since_rcnt_il_cat =  ifelse(
    is.na(mths_since_rcnt_il) == TRUE,
    "NONE",
    ifelse(
      mths_since_rcnt_il <= 12,
      "Less_1_Y",
      ifelse(
        mths_since_rcnt_il <= 24,
        "Less_2_Y",
        ifelse(
          mths_since_rcnt_il <= 36,
          "Less_3_Y",
          ifelse(mths_since_rcnt_il <= 48, "Less_4_Y", "More_4_Y")
        )
      )
    )
  )) %>% select(-mths_since_rcnt_il)

lc_data$mths_since_rcnt_il_cat <- as.factor(lc_data$mths_since_rcnt_il_cat)
ggplot(data = lc_data, mapping = aes(x=int_rate,y=mths_since_rcnt_il_cat))+geom_boxplot()

lc_data$mths_since_rcnt_il_cat <- as.numeric(lc_data$mths_since_rcnt_il_cat)
lc_data <-lc_data %>% 
  mutate(mths_since_last_major_derog_cat =  ifelse(
    is.na(mths_since_last_major_derog) == TRUE,
    "NONE",
    ifelse(
      mths_since_last_major_derog <= 12,
      "Less_1_Y",
      ifelse(
        mths_since_last_major_derog <= 24,
        "Less_2_Y",
        ifelse(
          mths_since_last_major_derog <= 36,
          "Less_3_Y",
          ifelse(mths_since_last_major_derog <= 48, "Less_4_Y", "More_4_Y")
        )
      )
    )
  )) %>% select(-mths_since_last_major_derog)

lc_data$mths_since_last_major_derog_cat <- as.factor(lc_data$mths_since_last_major_derog_cat)
ggplot(data = lc_data, mapping = aes(x=int_rate,y=mths_since_last_major_derog_cat))+geom_boxplot()

lc_data$mths_since_last_major_derog_cat <- as.numeric(lc_data$mths_since_last_major_derog_cat)

The variable initial_list_status identifies whether a loan was initially listed in the whole (W) or fractional (F) market. This variable could be useful so we can keep it and transform it to a factor and then to a numeric value, for the same purpose of compatibility with the XGBoost function.

lc_data$initial_list_status <- as.factor(lc_data$initial_list_status)
ggplot(data = lc_data, mapping = aes(x=int_rate,y=initial_list_status))+geom_boxplot()

lc_data$initial_list_status <- as.numeric(lc_data$initial_list_status)

Let’s check which columns still have null values

colSums(is.na(lc_data))
                      loan_amnt                            term                        int_rate 
                              0                               0                               0 
                     emp_length                  home_ownership             verification_status 
                              0                               0                               0 
                        purpose                      addr_state                     delinq_2yrs 
                              0                               0                               0 
               earliest_cr_line                  inq_last_6mths                        open_acc 
                              0                               0                               0 
                        pub_rec                       revol_bal                      revol_util 
                              0                               2                             428 
                      total_acc             initial_list_status      collections_12_mths_ex_med 
                              0                               0                              99 
                 acc_now_delinq                    tot_coll_amt                     tot_cur_bal 
                              0                           63132                           63132 
                    open_acc_6m                      open_il_6m                     open_il_12m 
                         779302                          779302                          779302 
                    open_il_24m                    total_bal_il                         il_util 
                         779302                          779302                          781784 
                    open_rv_12m                     open_rv_24m                      max_bal_bc 
                         779302                          779302                          779302 
                       all_util                total_rev_hi_lim                          inq_fi 
                         779302                           63132                          779302 
                    total_cu_tl                    inq_last_12m               annual_inc_merged 
                         779302                          779302                               0 
                     dti_merged           mths_since_delinq_cat      mths_since_last_record_cat 
                              0                               0                               0 
         mths_since_rcnt_il_cat mths_since_last_major_derog_cat 
                              0                               0 

The columns revol_bal and revol_util contain only few NA values, those values can’t be replaced with 0, then we filter the values which are not NAs.

lc_data <- lc_data %>% 
    filter(!(is.na(revol_bal))) %>% 
        filter(!(is.na(revol_util)))

Let’s check which columns still have null values:

names(which(colSums(is.na(lc_data)) > 0))
 [1] "collections_12_mths_ex_med" "tot_coll_amt"               "tot_cur_bal"                "open_acc_6m"               
 [5] "open_il_6m"                 "open_il_12m"                "open_il_24m"                "total_bal_il"              
 [9] "il_util"                    "open_rv_12m"                "open_rv_24m"                "max_bal_bc"                
[13] "all_util"                   "total_rev_hi_lim"           "inq_fi"                     "total_cu_tl"               
[17] "inq_last_12m"              

Replace null values with 0 where is possible

lc_data <-
  lc_data %>%
  mutate(open_acc_6m = ifelse(is.na(open_acc_6m) == TRUE, 0, open_acc_6m)) %>%
  mutate(tot_cur_bal = ifelse(is.na(tot_cur_bal) == TRUE, 0, tot_cur_bal)) %>%
  mutate(open_il_6m = ifelse(is.na(open_il_6m) == TRUE, 0, open_il_6m)) %>%
  mutate(open_il_12m = ifelse(is.na(open_il_12m) == TRUE, 0, open_il_12m)) %>%
  mutate(open_il_24m = ifelse(is.na(open_il_24m) == TRUE, 0, open_il_24m)) %>%
  mutate(total_bal_il = ifelse(is.na(total_bal_il) == TRUE, 0, total_bal_il)) %>%
  mutate(il_util = ifelse(is.na(il_util) == TRUE, 0, il_util)) %>%
  mutate(open_rv_12m = ifelse(is.na(open_rv_12m) == TRUE, 0, open_rv_12m)) %>%
  mutate(total_rev_hi_lim = ifelse(is.na(total_rev_hi_lim) == TRUE, 0, total_rev_hi_lim)) %>%
  mutate(max_bal_bc = ifelse(is.na(max_bal_bc) == TRUE, 0, max_bal_bc)) %>%
  mutate(all_util = ifelse(is.na(all_util) == TRUE, 0, all_util)) %>%
  mutate(inq_fi = ifelse(is.na(inq_fi) == TRUE, 0, inq_fi)) %>%
  mutate(total_cu_tl = ifelse(is.na(total_cu_tl) == TRUE, 0, total_cu_tl)) %>%
  mutate(inq_last_12m = ifelse(is.na(inq_last_12m) == TRUE, 0, inq_last_12m)) %>%
  mutate(open_rv_24m = ifelse(is.na(open_rv_24m) == TRUE, 0, open_rv_24m)) %>%
  mutate(tot_coll_amt = ifelse(is.na(tot_coll_amt)== TRUE,0, tot_coll_amt)) %>%
  mutate(collections_12_mths_ex_med = ifelse(is.na(collections_12_mths_ex_med)== TRUE,0, collections_12_mths_ex_med))

earliest_cr_line contains the month the borrower’s earliest reported credit line was opened. Even if this date consists only on month and year, still there are too many unique values. We could transform the dates in to a numerical value, by converting them from date into Unix Time. This unit measures time by the number of seconds that have elapsed since 00:00:00 UTC on 1 January 1970. Since this column doesn’t contain the day number, we take as a reference the first day of the month.

lc_data <- lc_data %>% 
    filter(!(is.na(earliest_cr_line)))

# function to replace dates with unix time
to_unix_time <- function(date) {
  tmp <- paste("01", date, sep="-")
  return (as.numeric(as.POSIXct(tmp, format="%d-%b-%Y", tz="UTC")))
}

# map dates to unix time
lc_data$earliest_cr_line <- apply(lc_data, 1, function(row) to_unix_time(row["earliest_cr_line"]))

# standardize them
#lc_data$earliest_cr_line <- scale(lc_data$earliest_cr_line)

Outliers Removal:

boxplot(lc_data$int_rate)

# Identify outliers using boxplot
outliers <- boxplot(lc_data$int_rate, plot = FALSE)$out
# Remove outliers from the dataset
lc_data_clean <- lc_data[!lc_data$int_rate %in% outliers, ]
summary(lc_data)
   loan_amnt          term        int_rate       emp_length    home_ownership  verification_status    purpose      
 Min.   :  500   Min.   :1.0   Min.   : 5.32   Min.   : 1.00   Min.   :1.000   Min.   :1.000       Min.   : 1.000  
 1st Qu.: 8000   1st Qu.:1.0   1st Qu.: 9.99   1st Qu.: 3.00   1st Qu.:1.000   1st Qu.:1.000       1st Qu.: 3.000  
 Median :13000   Median :1.0   Median :12.99   Median : 4.00   Median :2.000   Median :2.000       Median : 3.000  
 Mean   :14757   Mean   :1.3   Mean   :13.24   Mean   : 5.11   Mean   :1.901   Mean   :2.027       Mean   : 3.571  
 3rd Qu.:20000   3rd Qu.:2.0   3rd Qu.:16.20   3rd Qu.: 7.00   3rd Qu.:3.000   3rd Qu.:3.000       3rd Qu.: 3.000  
 Max.   :35000   Max.   :2.0   Max.   :28.99   Max.   :12.00   Max.   :3.000   Max.   :3.000       Max.   :14.000  
   addr_state     delinq_2yrs      earliest_cr_line     inq_last_6mths       open_acc        pub_rec       
 Min.   : 1.00   Min.   : 0.0000   Min.   :-820540800   Min.   : 0.0000   Min.   : 1.00   Min.   : 0.0000  
 1st Qu.:10.00   1st Qu.: 0.0000   1st Qu.: 770428800   1st Qu.: 0.0000   1st Qu.: 8.00   1st Qu.: 0.0000  
 Median :24.00   Median : 0.0000   Median : 936144000   Median : 0.0000   Median :11.00   Median : 0.0000  
 Mean   :24.14   Mean   : 0.3143   Mean   : 889273164   Mean   : 0.6947   Mean   :11.55   Mean   : 0.1954  
 3rd Qu.:37.00   3rd Qu.: 0.0000   3rd Qu.:1051747200   3rd Qu.: 1.0000   3rd Qu.:14.00   3rd Qu.: 0.0000  
 Max.   :51.00   Max.   :39.0000   Max.   :1351728000   Max.   :33.0000   Max.   :90.00   Max.   :63.0000  
   revol_bal         revol_util       total_acc      initial_list_status collections_12_mths_ex_med acc_now_delinq     
 Min.   :      0   Min.   :  0.00   Min.   :  1.00   Min.   :1.000       Min.   : 0.00000           Min.   : 0.000000  
 1st Qu.:   6450   1st Qu.: 37.70   1st Qu.: 17.00   1st Qu.:1.000       1st Qu.: 0.00000           1st Qu.: 0.000000  
 Median :  11881   Median : 56.00   Median : 24.00   Median :1.000       Median : 0.00000           Median : 0.000000  
 Mean   :  16934   Mean   : 55.05   Mean   : 25.27   Mean   :1.485       Mean   : 0.01448           Mean   : 0.005026  
 3rd Qu.:  20844   3rd Qu.: 73.50   3rd Qu.: 32.00   3rd Qu.:2.000       3rd Qu.: 0.00000           3rd Qu.: 0.000000  
 Max.   :2904836   Max.   :892.30   Max.   :169.00   Max.   :2.000       Max.   :20.00000           Max.   :14.000000  
  tot_coll_amt      tot_cur_bal       open_acc_6m         open_il_6m        open_il_12m        open_il_24m      
 Min.   :      0   Min.   :      0   Min.   : 0.00000   Min.   : 0.00000   Min.   : 0.00000   Min.   : 0.00000  
 1st Qu.:      0   1st Qu.:  23195   1st Qu.: 0.00000   1st Qu.: 0.00000   1st Qu.: 0.00000   1st Qu.: 0.00000  
 Median :      0   Median :  65402   Median : 0.00000   Median : 0.00000   Median : 0.00000   Median : 0.00000  
 Mean   :    210   Mean   : 128461   Mean   : 0.02641   Mean   : 0.06982   Mean   : 0.01816   Mean   : 0.03991  
 3rd Qu.:      0   3rd Qu.: 195864   3rd Qu.: 0.00000   3rd Qu.: 0.00000   3rd Qu.: 0.00000   3rd Qu.: 0.00000  
 Max.   :9152545   Max.   :8000078   Max.   :14.00000   Max.   :33.00000   Max.   :12.00000   Max.   :19.00000  
  total_bal_il       il_util         open_rv_12m        open_rv_24m         max_bal_bc         all_util      
 Min.   :     0   Min.   :  0.000   Min.   : 0.00000   Min.   : 0.00000   Min.   :    0.0   Min.   :  0.000  
 1st Qu.:     0   1st Qu.:  0.000   1st Qu.: 0.00000   1st Qu.: 0.00000   1st Qu.:    0.0   1st Qu.:  0.000  
 Median :     0   Median :  0.000   Median : 0.00000   Median : 0.00000   Median :    0.0   Median :  0.000  
 Mean   :   872   Mean   :  1.489   Mean   : 0.03316   Mean   : 0.07114   Mean   :  140.8   Mean   :  1.456  
 3rd Qu.:     0   3rd Qu.:  0.000   3rd Qu.: 0.00000   3rd Qu.: 0.00000   3rd Qu.:    0.0   3rd Qu.:  0.000  
 Max.   :878459   Max.   :223.300   Max.   :22.00000   Max.   :43.00000   Max.   :83047.0   Max.   :151.400  
 total_rev_hi_lim      inq_fi          total_cu_tl        inq_last_12m      annual_inc_merged   dti_merged   
 Min.   :      0   Min.   : 0.00000   Min.   : 0.00000   Min.   :-4.00000   Min.   :   1896   Min.   : 0.00  
 1st Qu.:  11700   1st Qu.: 0.00000   1st Qu.: 0.00000   1st Qu.: 0.00000   1st Qu.:  45000   1st Qu.:11.91  
 Median :  21800   Median : 0.00000   Median : 0.00000   Median : 0.00000   Median :  65000   Median :17.66  
 Mean   :  29564   Mean   : 0.02262   Mean   : 0.03668   Mean   : 0.04733   Mean   :  75038   Mean   :18.13  
 3rd Qu.:  37900   3rd Qu.: 0.00000   3rd Qu.: 0.00000   3rd Qu.: 0.00000   3rd Qu.:  90000   3rd Qu.:23.94  
 Max.   :9999999   Max.   :16.00000   Max.   :35.00000   Max.   :32.00000   Max.   :9500000   Max.   :43.86  
 mths_since_delinq_cat mths_since_last_record_cat mths_since_rcnt_il_cat mths_since_last_major_derog_cat
 Min.   :1.000         Min.   :1.000              Min.   :1.000          Min.   :1.000                  
 1st Qu.:3.000         1st Qu.:6.000              1st Qu.:6.000          1st Qu.:6.000                  
 Median :6.000         Median :6.000              Median :6.000          Median :6.000                  
 Mean   :4.576         Mean   :5.779              Mean   :5.906          Mean   :5.435                  
 3rd Qu.:6.000         3rd Qu.:6.000              3rd Qu.:6.000          3rd Qu.:6.000                  
 Max.   :6.000         Max.   :6.000              Max.   :6.000          Max.   :6.000                  

Learning Algorithms

# create indices for splitting (80% train, 20% test)
train_indices <- createDataPartition(lc_data$int_rate, p = 0.8, list = FALSE)

# create training and testing datasets
train_data <- lc_data[train_indices, ]
test_data <- lc_data[-train_indices, ]
#### Linear Regression ####

lm.fit <- lm(int_rate ~ ., data = train_data)

# make predictions on the training and testing data
lm.train_predictions <- predict(lm.fit, newdata = train_data)
lm.test_predictions <- predict(lm.fit, newdata = test_data)

# calculate Mean Squared Error (MSE) for training and testing
lm.train_mse <- mean((lm.train_predictions - train_data$int_rate)^2)
lm.test_mse <- mean((lm.test_predictions - test_data$int_rate)^2)

# calculate Root Mean Squared Error (RMSE) for training and testing
lm.train_rmse <- sqrt(lm.train_mse)
lm.test_rmse <- sqrt(lm.test_mse)

# calculate Mean Absolute Error (MAE) for training and testing
lm.train_mae <- mean(abs(lm.train_predictions - train_data$int_rate))
lm.test_mae <- mean(abs(lm.test_predictions - test_data$int_rate))

# calculate R-squared (R²) for training and testing
lm.train_r2 <- 1 - (sum((train_data$int_rate - lm.train_predictions)^2) / sum((train_data$int_rate - mean(train_data$int_rate))^2))
lm.test_r2 <- 1 - (sum((test_data$int_rate - lm.test_predictions)^2) / sum((test_data$int_rate - mean(test_data$int_rate))^2))

# display the metrics
cat("Training MSE:", lm.train_mse, "\n")
Training MSE: 10.69206 
cat("Testing MSE:", lm.test_mse, "\n")
Testing MSE: 11.07953 
cat("Training RMSE:", lm.train_rmse, "\n")
Training RMSE: 3.269871 
cat("Testing RMSE:", lm.test_rmse, "\n")
Testing RMSE: 3.328593 
cat("Training MAE:", lm.train_mae, "\n")
Training MAE: 2.591219 
cat("Testing MAE:", lm.test_mae, "\n")
Testing MAE: 2.591901 
cat("Training R-squared (R²):", lm.train_r2, "\n")
Training R-squared (R²): 0.4432613 
cat("Testing R-squared (R²):", lm.test_r2, "\n")
Testing R-squared (R²): 0.4228202 

Lasso It standardizes data automatically

lasso.predictors_train <- model.matrix(int_rate ~ ., train_data)[,-1]
lasso.target_train <- train_data$int_rate
lasso.predictors_test <- model.matrix(int_rate ~ ., test_data)[,-1]
lasso.target_test <- test_data$int_rate

lasso.fit <- glmnet(lasso.predictors_train, lasso.target_train, alpha = 1)

plot(lasso.fit, label=TRUE)


# make predictions on the training and testing data
lasso.train_predictions <- predict(lasso.fit, newdata = train_data, newx = lasso.predictors_train)
lasso.test_predictions <- predict(lasso.fit, newdata = test_data, newx = lasso.predictors_train)

# calculate Mean Squared Error (MSE) for training and testing
lasso.train_mse <- mean((lasso.train_predictions - train_data$int_rate)^2)
lasso.test_mse <- mean((lasso.test_predictions - test_data$int_rate)^2)
Warning: longer object length is not a multiple of shorter object length
# calculate Root Mean Squared Error (RMSE) for training and testing
lasso.train_rmse <- sqrt(lasso.train_mse)
lasso.test_rmse <- sqrt(lasso.test_mse)

# calculate Mean Absolute Error (MAE) for training and testing
lasso.train_mae <- mean(abs(lasso.train_predictions - train_data$int_rate))
lasso.test_mae <- mean(abs(lasso.test_predictions - test_data$int_rate))
Warning: longer object length is not a multiple of shorter object length
# calculate R-squared (R²) for training and testing
lasso.train_r2 <- 1 - (sum((train_data$int_rate - lasso.train_predictions)^2) / sum((train_data$int_rate - mean(train_data$int_rate))^2))
lasso.test_r2 <- 1 - (sum((test_data$int_rate - lasso.test_predictions)^2) / sum((test_data$int_rate - mean(test_data$int_rate))^2))
Warning: longer object length is not a multiple of shorter object length
# display the metrics
cat("Training MSE:", lasso.train_mse, "\n")
Training MSE: 12.01177 
cat("Testing MSE:", lasso.test_mse, "\n")
Testing MSE: 25.14051 
cat("Training RMSE:", lasso.train_rmse, "\n")
Training RMSE: 3.4658 
cat("Testing RMSE:", lasso.test_rmse, "\n")
Testing RMSE: 5.014031 
cat("Training MAE:", lasso.train_mae, "\n")
Training MAE: 2.754045 
cat("Testing MAE:", lasso.test_mae, "\n")
Testing MAE: 4.012612 
cat("Training R-squared (R²):", lasso.train_r2, "\n")
Training R-squared (R²): -46.53467 
cat("Testing R-squared (R²):", lasso.test_r2, "\n")
Testing R-squared (R²): -397.1465 

Ridge It standardizes data automatically

ridge.predictors_train <- model.matrix(int_rate ~ ., train_data)[,-1]
ridge.target_train <- train_data$int_rate
ridge.predictors_test <- model.matrix(int_rate ~ ., test_data)[,-1]
ridge.target_test <- test_data$int_rate

ridge.fit <- glmnet(ridge.predictors_train, ridge.target_train, alpha = 0)

plot(ridge.fit, label=TRUE, xlab = "L2 Norm")


# make predictions on the training and testing data
ridge.train_predictions <- predict(ridge.fit, newdata = train_data, newx = ridge.predictors_train)
ridge.test_predictions <- predict(ridge.fit, newdata = test_data, newx = ridge.predictors_train)

# calculate Mean Squared Error (MSE) for training and testing
ridge.train_mse <- mean((ridge.train_predictions - train_data$int_rate)^2)
ridge.test_mse <- mean((ridge.test_predictions - test_data$int_rate)^2)
Warning: longer object length is not a multiple of shorter object length
# calculate Root Mean Squared Error (RMSE) for training and testing
ridge.train_rmse <- sqrt(ridge.train_mse)
ridge.test_rmse <- sqrt(ridge.test_mse)

# calculate Mean Absolute Error (MAE) for training and testing
ridge.train_mae <- mean(abs(ridge.train_predictions - train_data$int_rate))
ridge.test_mae <- mean(abs(ridge.test_predictions - test_data$int_rate))
Warning: longer object length is not a multiple of shorter object length
# calculate R-squared (R²) for training and testing
ridge.train_r2 <- 1 - (sum((train_data$int_rate - ridge.train_predictions)^2) / sum((train_data$int_rate - mean(train_data$int_rate))^2))
ridge.test_r2 <- 1 - (sum((test_data$int_rate - ridge.test_predictions)^2) / sum((test_data$int_rate - mean(test_data$int_rate))^2))
Warning: longer object length is not a multiple of shorter object length
# display the metrics
cat("Training MSE:", ridge.train_mse, "\n")
Training MSE: 15.24403 
cat("Testing MSE:", ridge.test_mse, "\n")
Testing MSE: 21.3636 
cat("Training RMSE:", ridge.train_rmse, "\n")
Training RMSE: 3.90436 
cat("Testing RMSE:", ridge.test_rmse, "\n")
Testing RMSE: 4.622077 
cat("Training MAE:", ridge.train_mae, "\n")
Training MAE: 3.102204 
cat("Testing MAE:", ridge.test_mae, "\n")
Testing MAE: 3.703659 
cat("Training R-squared (R²):", ridge.train_r2, "\n")
Training R-squared (R²): -78.3761 
cat("Testing R-squared (R²):", ridge.test_r2, "\n")
Testing R-squared (R²): -444.1739 

K fold using K=5:

# define the number of folds for cross-validation
num_folds <- 5
folds <- createFolds(train_data$int_rate, k = num_folds, list = TRUE)

K fold using K=5 and linear regression:

#### Linear Regresion applying Cross Validation with k=5  ####

# initialize lists to store models and their results
lm.k5.models <- list()
lm.k5.results <- data.frame()

# perform k-fold cross-validation
for(i in seq_along(folds)) {
  # split the data into training and testing for the current fold
  train_indices <- folds[[i]]
  test_indices <- setdiff(seq_len(nrow(train_data)), train_indices)
  
  train_data_fold <- train_data[train_indices, ]
  test_data_fold <- train_data[test_indices, ]
  
  # fit the model on the training fold
  lm.k5 <- lm(int_rate ~ ., data = train_data_fold)
  lm.k5.models[[i]] <- lm.k5  # Store the model
  
  # make predictions on the training and testing fold
  lm.k5.train_predictions <- predict(lm.k5, newdata = train_data_fold)
  lm.k5.test_predictions <- predict(lm.k5, newdata = test_data_fold)
  
  # calculate metrics for training fold
  lm.k5.train_mse <- mean((lm.k5.train_predictions - train_data_fold$int_rate)^2)
  lm.k5.train_rmse <- sqrt(lm.k5.train_mse)
  lm.k5.train_mae <- mean(abs(lm.k5.train_predictions - train_data_fold$int_rate))
  lm.k5.train_r2 <- summary(lm.k5)$r.squared
  
  # calculate metrics for testing fold
  lm.k5.test_mse <- mean((lm.k5.test_predictions - test_data_fold$int_rate)^2)
  lm.k5.test_rmse <- sqrt(lm.k5.test_mse)
  lm.k5.test_mae <- mean(abs(lm.k5.test_predictions - test_data_fold$int_rate))
  lm.k5.test_r2 <- 1 - (sum((test_data_fold$int_rate - lm.k5.test_predictions)^2) / sum((test_data_fold$int_rate - mean(test_data_fold$int_rate))^2))
  
  # store metrics in the results dataframe
  lm.k5.results <- rbind(lm.k5.results, data.frame(
    Fold = i,
    Train_MSE = lm.k5.train_mse, Test_MSE = lm.k5.test_mse,
    Train_RMSE = lm.k5.train_rmse, Test_RMSE = lm.k5.test_rmse,
    Train_MAE = lm.k5.train_mae, Test_MAE = lm.k5.test_mae,
    Train_R2 = lm.k5.train_r2, Test_R2 = lm.k5.test_r2
  ))
}

# display the models and their metrics
print(lm.k5.models)
[[1]]

Call:
lm(formula = int_rate ~ ., data = train_data_fold)

Coefficients:
                    (Intercept)                        loan_amnt                             term  
                      1.910e+00                        3.233e-05                        3.920e+00  
                     emp_length                   home_ownership              verification_status  
                      1.263e-02                        2.509e-01                        7.581e-01  
                        purpose                       addr_state                      delinq_2yrs  
                      3.386e-01                        6.130e-05                        3.605e-02  
               earliest_cr_line                   inq_last_6mths                         open_acc  
                      1.871e-09                        9.649e-01                        6.637e-02  
                        pub_rec                        revol_bal                       revol_util  
                      3.463e-01                        3.865e-06                        4.248e-02  
                      total_acc              initial_list_status       collections_12_mths_ex_med  
                     -3.603e-02                       -1.014e+00                        2.191e-01  
                 acc_now_delinq                     tot_coll_amt                      tot_cur_bal  
                      1.026e+00                        2.246e-05                       -1.206e-06  
                    open_acc_6m                       open_il_6m                      open_il_12m  
                     -1.330e-02                       -1.596e-01                        7.716e-01  
                    open_il_24m                     total_bal_il                          il_util  
                      4.741e-02                        7.607e-07                        9.710e-03  
                    open_rv_12m                      open_rv_24m                       max_bal_bc  
                      1.586e-01                        7.933e-02                       -3.460e-05  
                       all_util                 total_rev_hi_lim                           inq_fi  
                     -4.508e-03                       -1.646e-05                        6.036e-02  
                    total_cu_tl                     inq_last_12m                annual_inc_merged  
                     -4.045e-02                        7.753e-02                       -3.377e-06  
                     dti_merged            mths_since_delinq_cat       mths_since_last_record_cat  
                      5.160e-02                       -2.022e-01                       -1.924e-01  
         mths_since_rcnt_il_cat  mths_since_last_major_derog_cat  
                      3.467e-01                       -1.427e-01  


[[2]]

Call:
lm(formula = int_rate ~ ., data = train_data_fold)

Coefficients:
                    (Intercept)                        loan_amnt                             term  
                      2.041e+00                        2.416e-05                        3.944e+00  
                     emp_length                   home_ownership              verification_status  
                      1.658e-02                        2.442e-01                        7.451e-01  
                        purpose                       addr_state                      delinq_2yrs  
                      3.440e-01                       -4.483e-04                        3.113e-02  
               earliest_cr_line                   inq_last_6mths                         open_acc  
                      1.998e-09                        9.973e-01                        4.845e-02  
                        pub_rec                        revol_bal                       revol_util  
                      3.922e-01                       -6.678e-06                        4.788e-02  
                      total_acc              initial_list_status       collections_12_mths_ex_med  
                     -3.298e-02                       -1.071e+00                        3.099e-01  
                 acc_now_delinq                     tot_coll_amt                      tot_cur_bal  
                      1.257e+00                        2.136e-05                       -1.509e-06  
                    open_acc_6m                       open_il_6m                      open_il_12m  
                      1.282e-01                       -1.558e-01                        6.939e-01  
                    open_il_24m                     total_bal_il                          il_util  
                      3.900e-02                        2.395e-06                        3.730e-03  
                    open_rv_12m                      open_rv_24m                       max_bal_bc  
                      1.319e-01                        6.035e-02                       -7.830e-05  
                       all_util                 total_rev_hi_lim                           inq_fi  
                      1.620e-03                       -3.147e-06                        9.833e-02  
                    total_cu_tl                     inq_last_12m                annual_inc_merged  
                     -7.565e-02                        6.516e-02                       -3.608e-06  
                     dti_merged            mths_since_delinq_cat       mths_since_last_record_cat  
                      5.144e-02                       -2.073e-01                       -1.765e-01  
         mths_since_rcnt_il_cat  mths_since_last_major_derog_cat  
                      2.746e-01                       -1.533e-01  


[[3]]

Call:
lm(formula = int_rate ~ ., data = train_data_fold)

Coefficients:
                    (Intercept)                        loan_amnt                             term  
                      1.706e+00                        3.616e-05                        3.867e+00  
                     emp_length                   home_ownership              verification_status  
                      1.486e-02                        2.624e-01                        7.410e-01  
                        purpose                       addr_state                      delinq_2yrs  
                      3.365e-01                        6.215e-04                        4.298e-02  
               earliest_cr_line                   inq_last_6mths                         open_acc  
                      1.890e-09                        9.889e-01                        6.379e-02  
                        pub_rec                        revol_bal                       revol_util  
                      4.020e-01                        5.529e-06                        4.188e-02  
                      total_acc              initial_list_status       collections_12_mths_ex_med  
                     -3.451e-02                       -1.025e+00                        3.280e-01  
                 acc_now_delinq                     tot_coll_amt                      tot_cur_bal  
                      1.464e+00                        3.626e-05                       -1.005e-06  
                    open_acc_6m                       open_il_6m                      open_il_12m  
                     -6.804e-03                       -1.613e-01                        7.425e-01  
                    open_il_24m                     total_bal_il                          il_util  
                      6.193e-02                        2.215e-06                        5.888e-03  
                    open_rv_12m                      open_rv_24m                       max_bal_bc  
                      2.849e-01                        4.946e-03                       -4.102e-05  
                       all_util                 total_rev_hi_lim                           inq_fi  
                      1.527e-04                       -1.706e-05                        2.065e-03  
                    total_cu_tl                     inq_last_12m                annual_inc_merged  
                     -2.524e-02                        9.353e-02                       -4.838e-06  
                     dti_merged            mths_since_delinq_cat       mths_since_last_record_cat  
                      5.081e-02                       -1.950e-01                       -1.511e-01  
         mths_since_rcnt_il_cat  mths_since_last_major_derog_cat  
                      3.557e-01                       -1.507e-01  


[[4]]

Call:
lm(formula = int_rate ~ ., data = train_data_fold)

Coefficients:
                    (Intercept)                        loan_amnt                             term  
                      1.893e+00                        2.584e-05                        3.910e+00  
                     emp_length                   home_ownership              verification_status  
                      1.312e-02                        2.583e-01                        7.449e-01  
                        purpose                       addr_state                      delinq_2yrs  
                      3.440e-01                        2.482e-05                        4.644e-02  
               earliest_cr_line                   inq_last_6mths                         open_acc  
                      1.989e-09                        9.995e-01                        4.512e-02  
                        pub_rec                        revol_bal                       revol_util  
                      4.346e-01                       -3.946e-06                        4.732e-02  
                      total_acc              initial_list_status       collections_12_mths_ex_med  
                     -3.150e-02                       -1.090e+00                        3.695e-01  
                 acc_now_delinq                     tot_coll_amt                      tot_cur_bal  
                      1.287e+00                        3.196e-05                       -1.526e-06  
                    open_acc_6m                       open_il_6m                      open_il_12m  
                     -6.852e-02                       -1.189e-01                        8.625e-01  
                    open_il_24m                     total_bal_il                          il_util  
                      2.418e-02                        1.271e-07                        4.949e-03  
                    open_rv_12m                      open_rv_24m                       max_bal_bc  
                      2.133e-01                        6.556e-02                       -6.542e-05  
                       all_util                 total_rev_hi_lim                           inq_fi  
                      9.265e-04                       -3.429e-06                        6.894e-02  
                    total_cu_tl                     inq_last_12m                annual_inc_merged  
                     -6.816e-02                        6.176e-02                       -4.018e-06  
                     dti_merged            mths_since_delinq_cat       mths_since_last_record_cat  
                      5.046e-02                       -2.031e-01                       -1.855e-01  
         mths_since_rcnt_il_cat  mths_since_last_major_derog_cat  
                      3.063e-01                       -1.426e-01  


[[5]]

Call:
lm(formula = int_rate ~ ., data = train_data_fold)

Coefficients:
                    (Intercept)                        loan_amnt                             term  
                      1.990e+00                        3.206e-05                        3.892e+00  
                     emp_length                   home_ownership              verification_status  
                      1.746e-02                        2.751e-01                        7.288e-01  
                        purpose                       addr_state                      delinq_2yrs  
                      3.330e-01                        6.527e-04                        2.890e-02  
               earliest_cr_line                   inq_last_6mths                         open_acc  
                      1.894e-09                        9.916e-01                        6.403e-02  
                        pub_rec                        revol_bal                       revol_util  
                      4.223e-01                        7.105e-06                        4.250e-02  
                      total_acc              initial_list_status       collections_12_mths_ex_med  
                     -3.643e-02                       -9.932e-01                        2.520e-01  
                 acc_now_delinq                     tot_coll_amt                      tot_cur_bal  
                      1.367e+00                        2.487e-05                       -1.039e-06  
                    open_acc_6m                       open_il_6m                      open_il_12m  
                      9.287e-02                       -1.552e-01                        6.460e-01  
                    open_il_24m                     total_bal_il                          il_util  
                      4.842e-02                        3.241e-06                        5.126e-03  
                    open_rv_12m                      open_rv_24m                       max_bal_bc  
                      1.508e-01                        8.182e-02                       -6.169e-05  
                       all_util                 total_rev_hi_lim                           inq_fi  
                     -6.585e-04                       -1.709e-05                       -3.103e-02  
                    total_cu_tl                     inq_last_12m                annual_inc_merged  
                     -5.278e-02                        9.582e-02                       -3.462e-06  
                     dti_merged            mths_since_delinq_cat       mths_since_last_record_cat  
                      5.261e-02                       -2.006e-01                       -1.559e-01  
         mths_since_rcnt_il_cat  mths_since_last_major_derog_cat  
                      2.809e-01                       -1.417e-01  
print(lm.k5.results)
plot_metric <- function(results_long, metric) {
    # adjust the variable names based on the metric
    variables <- if (metric == "OOB") {
        "OOB_Error"
    } else {
        c(paste0('Train_', metric), paste0('Test_', metric))
    }
    title <- if (metric == "OOB") {
        paste0(metric, ' per Fold')
    } else {
        paste0('Train vs Test ', metric, ' per Fold')
    }
    
    ggplot(results_long[results_long$variable %in% variables, ],
           aes(x = Fold, y = value, color = variable)) +
    geom_line() +
    geom_point() +
    theme_minimal() +
    labs(title = title,
         x = 'Fold',
         y = metric)
}
# reshape data for plotting
lm.k5.results_long <- melt(lm.k5.results, id.vars = 'Fold')

# plot for each metric
p1 <- plot_metric(lm.k5.results_long, 'MSE')
p2 <- plot_metric(lm.k5.results_long, 'RMSE')
p3 <- plot_metric(lm.k5.results_long, 'MAE')
p4 <- plot_metric(lm.k5.results_long, 'R2')

# arrange the plots in a 2x2 grid
grid.arrange(p1, p2, p3, p4, ncol = 2, nrow = 2)


plot(p1)

plot(p2)

plot(p3)

plot(p4)

K fold using K=5 and Random Forest:

#### Random Forest applying Cross Validation with k=5  ####

# initialize lists to store models and their results
rf.k5.models <- list()
rf.k5.results <- data.frame()

# perform k-fold cross-validation
for(i in seq_along(folds)) {
  # split the data into training and testing for the current fold
  train_indices <- folds[[i]]
  test_indices <- setdiff(seq_len(nrow(train_data)), train_indices)
  
  train_data_fold <- train_data[train_indices, ]
  test_data_fold <- train_data[test_indices, ]
  
  # fit the model on the training fold
  rf.k5 <- ranger(formula = int_rate ~ ., data = train_data, num.trees = 500, verbose=TRUE, importance = "impurity", oob.error = TRUE)
  rf.k5.models[[i]] <- rf.k5  # Store the model
  
  # make predictions on the training and testing fold
  rf.k5.train_predictions <- predict(rf.k5, data = train_data_fold)$predictions
  rf.k5.test_predictions <- predict(rf.k5, data = test_data_fold)$predictions
  
  # calculate metrics for training fold
  rf.k5.train_mse <- mean((rf.k5.train_predictions - train_data_fold$int_rate)^2)
  rf.k5.train_rmse <- sqrt(rf.k5.train_mse)
  rf.k5.train_mae <- mean(abs(rf.k5.train_predictions - train_data_fold$int_rate))
  rf.k5.oob_error <- rf.k5$prediction.error
  
  # calculate metrics for testing fold
  rf.k5.test_mse <- mean((rf.k5.test_predictions - test_data_fold$int_rate)^2)
  rf.k5.test_rmse <- sqrt(rf.k5.test_mse)
  rf.k5.test_mae <- mean(abs(rf.k5.test_predictions - test_data_fold$int_rate))
  rf.k5.test_r2 <- 1 - (sum((test_data_fold$int_rate - rf.k5.test_predictions)^2) / sum((test_data_fold$int_rate - mean(test_data_fold$int_rate))^2))
  
  # store metrics in the results dataframe
  rf.k5.results <- rbind(rf.k5.results, data.frame(
    Fold = i,
    Train_MSE = rf.k5.train_mse, Test_MSE = rf.k5.test_mse,
    Train_RMSE = rf.k5.train_rmse, Test_RMSE = rf.k5.test_rmse,
    Train_MAE = rf.k5.train_mae, Test_MAE = rf.k5.test_mae,
    OOB_Error = rf.k5.oob_error
  ))
}
Growing trees.. Progress: 16%. Estimated remaining time: 2 minutes, 47 seconds.
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Growing trees.. Progress: 64%. Estimated remaining time: 1 minute, 12 seconds.
Growing trees.. Progress: 80%. Estimated remaining time: 38 seconds.
Growing trees.. Progress: 96%. Estimated remaining time: 7 seconds.
# display the models and their metrics
print(rf.k5.models)
[[1]]
Ranger result

Call:
 ranger(formula = int_rate ~ ., data = train_data, num.trees = 500,      verbose = TRUE, importance = "impurity", oob.error = TRUE) 

Type:                             Regression 
Number of trees:                  500 
Sample size:                      638392 
Number of independent variables:  40 
Mtry:                             6 
Target node size:                 5 
Variable importance mode:         impurity 
Splitrule:                        variance 
OOB prediction error (MSE):       8.751032 
R squared (OOB):                  0.5443319 

[[2]]
Ranger result

Call:
 ranger(formula = int_rate ~ ., data = train_data, num.trees = 500,      verbose = TRUE, importance = "impurity", oob.error = TRUE) 

Type:                             Regression 
Number of trees:                  500 
Sample size:                      638392 
Number of independent variables:  40 
Mtry:                             6 
Target node size:                 5 
Variable importance mode:         impurity 
Splitrule:                        variance 
OOB prediction error (MSE):       8.746286 
R squared (OOB):                  0.544579 

[[3]]
Ranger result

Call:
 ranger(formula = int_rate ~ ., data = train_data, num.trees = 500,      verbose = TRUE, importance = "impurity", oob.error = TRUE) 

Type:                             Regression 
Number of trees:                  500 
Sample size:                      638392 
Number of independent variables:  40 
Mtry:                             6 
Target node size:                 5 
Variable importance mode:         impurity 
Splitrule:                        variance 
OOB prediction error (MSE):       8.749652 
R squared (OOB):                  0.5444038 

[[4]]
Ranger result

Call:
 ranger(formula = int_rate ~ ., data = train_data, num.trees = 500,      verbose = TRUE, importance = "impurity", oob.error = TRUE) 

Type:                             Regression 
Number of trees:                  500 
Sample size:                      638392 
Number of independent variables:  40 
Mtry:                             6 
Target node size:                 5 
Variable importance mode:         impurity 
Splitrule:                        variance 
OOB prediction error (MSE):       8.746698 
R squared (OOB):                  0.5445576 

[[5]]
Ranger result

Call:
 ranger(formula = int_rate ~ ., data = train_data, num.trees = 500,      verbose = TRUE, importance = "impurity", oob.error = TRUE) 

Type:                             Regression 
Number of trees:                  500 
Sample size:                      638392 
Number of independent variables:  40 
Mtry:                             6 
Target node size:                 5 
Variable importance mode:         impurity 
Splitrule:                        variance 
OOB prediction error (MSE):       8.747217 
R squared (OOB):                  0.5445306 
print(rf.k5.results)
# reshape data for plotting
rf.k5.results_long <- melt(rf.k5.results, id.vars = 'Fold')

# plot for each metric
p1 <- plot_metric(rf.k5.results_long, 'MSE')
p2 <- plot_metric(rf.k5.results_long, 'RMSE')
p3 <- plot_metric(rf.k5.results_long, 'MAE')
p4 <- plot_metric(rf.k5.results_long, 'OOB')

# arrange the plots in a 2x2 grid
grid.arrange(p1, p2, p3, p4, ncol = 2, nrow = 2)


plot(p1)

plot(p2)

plot(p3)

plot(p4)

K fold using K=5 and Boosting:

#### Boosting applying Cross Validation with k=5  ####

# initialize lists to store models and their results
xgb.k5.models <- list()
xgb.k5.results <- data.frame()

# perform k-fold cross-validation
for(i in seq_along(folds)) {
  # split the data into training and testing for the current fold
  train_indices <- folds[[i]]
  test_indices <- setdiff(seq_len(nrow(train_data)), train_indices)
  
  train_data_fold <- train_data[train_indices, ]
  test_data_fold <- train_data[test_indices, ]
  
  # prepare data for xgboost
  xgb.y_train_fold <- train_data_fold$int_rate
  xgb.X_train_fold <- as.matrix(train_data_fold[, -which(names(train_data_fold) == 'int_rate')])
  
  xgb.y_test_fold <- test_data_fold$int_rate
  xgb.X_test_fold <- as.matrix(test_data_fold[, -which(names(test_data_fold) == 'int_rate')])
  
  # fit the xgboost model on the training fold
  xgb.k5 <- xgboost(
    data = xgb.X_train_fold,
    label = xgb.y_train_fold,
    nrounds = 100,
    verbose = 0
  )
  xgb.k5.models[[i]] <- xgb.k5  # store the model
  
  # make predictions on the training fold
  xgb.k5.train_predictions <- predict(xgb.k5, newdata = xgb.X_train_fold)
  # make predictions on the testing fold
  xgb.k5.test_predictions <- predict(xgb.k5, newdata = xgb.X_test_fold)
  
  # calculate metrics for training fold
  xgb.k5.train_mse <- mean((xgb.k5.train_predictions - train_data_fold$int_rate)^2)
  xgb.k5.train_rmse <- sqrt(xgb.k5.train_mse)
  xgb.k5.train_mae <- mean(abs(xgb.k5.train_predictions - train_data_fold$int_rate))
  xgb.k5.train_r2 <- 1 - (sum((xgb.y_train_fold - xgb.k5.train_predictions)^2) / sum((xgb.y_train_fold - mean(xgb.y_train_fold))^2))

  # calculate metrics for testing fold
  xgb.k5.test_mse <- mean((xgb.k5.test_predictions - xgb.y_test_fold)^2)
  xgb.k5.test_rmse <- sqrt(xgb.k5.test_mse)
  xgb.k5.test_mae <- mean(abs(xgb.k5.test_predictions - xgb.y_test_fold))
  xgb.k5.test_r2 <- 1 - (sum((xgb.y_test_fold - xgb.k5.test_predictions)^2) / sum((xgb.y_test_fold - mean(xgb.y_test_fold))^2))  
  
  # store metrics in the results dataframe
  xgb.k5.results <- rbind(xgb.k5.results, data.frame(
    Fold = i,
    Train_MSE = xgb.k5.train_mse, Test_MSE = xgb.k5.test_mse,
    Train_RMSE = xgb.k5.train_rmse, Test_RMSE = xgb.k5.test_rmse,
    Train_MAE = xgb.k5.train_mae, Test_MAE = xgb.k5.test_mae,
    Train_R2 = xgb.k5.train_r2, Test_R2 = xgb.k5.test_r2
  ))
}

# display the models and their metrics
print(xgb.k5.models)
[[1]]
##### xgb.Booster
raw: 450.2 Kb 
call:
  xgb.train(params = params, data = dtrain, nrounds = nrounds, 
    watchlist = watchlist, verbose = verbose, print_every_n = print_every_n, 
    early_stopping_rounds = early_stopping_rounds, maximize = maximize, 
    save_period = save_period, save_name = save_name, xgb_model = xgb_model, 
    callbacks = callbacks)
params (as set within xgb.train):
  validate_parameters = "TRUE"
xgb.attributes:
  niter
callbacks:
  cb.evaluation.log()
# of features: 40 
niter: 100
nfeatures : 40 
evaluation_log:

[[2]]
##### xgb.Booster
raw: 442.2 Kb 
call:
  xgb.train(params = params, data = dtrain, nrounds = nrounds, 
    watchlist = watchlist, verbose = verbose, print_every_n = print_every_n, 
    early_stopping_rounds = early_stopping_rounds, maximize = maximize, 
    save_period = save_period, save_name = save_name, xgb_model = xgb_model, 
    callbacks = callbacks)
params (as set within xgb.train):
  validate_parameters = "TRUE"
xgb.attributes:
  niter
callbacks:
  cb.evaluation.log()
# of features: 40 
niter: 100
nfeatures : 40 
evaluation_log:

[[3]]
##### xgb.Booster
raw: 443.5 Kb 
call:
  xgb.train(params = params, data = dtrain, nrounds = nrounds, 
    watchlist = watchlist, verbose = verbose, print_every_n = print_every_n, 
    early_stopping_rounds = early_stopping_rounds, maximize = maximize, 
    save_period = save_period, save_name = save_name, xgb_model = xgb_model, 
    callbacks = callbacks)
params (as set within xgb.train):
  validate_parameters = "TRUE"
xgb.attributes:
  niter
callbacks:
  cb.evaluation.log()
# of features: 40 
niter: 100
nfeatures : 40 
evaluation_log:

[[4]]
##### xgb.Booster
raw: 447.1 Kb 
call:
  xgb.train(params = params, data = dtrain, nrounds = nrounds, 
    watchlist = watchlist, verbose = verbose, print_every_n = print_every_n, 
    early_stopping_rounds = early_stopping_rounds, maximize = maximize, 
    save_period = save_period, save_name = save_name, xgb_model = xgb_model, 
    callbacks = callbacks)
params (as set within xgb.train):
  validate_parameters = "TRUE"
xgb.attributes:
  niter
callbacks:
  cb.evaluation.log()
# of features: 40 
niter: 100
nfeatures : 40 
evaluation_log:

[[5]]
##### xgb.Booster
raw: 451 Kb 
call:
  xgb.train(params = params, data = dtrain, nrounds = nrounds, 
    watchlist = watchlist, verbose = verbose, print_every_n = print_every_n, 
    early_stopping_rounds = early_stopping_rounds, maximize = maximize, 
    save_period = save_period, save_name = save_name, xgb_model = xgb_model, 
    callbacks = callbacks)
params (as set within xgb.train):
  validate_parameters = "TRUE"
xgb.attributes:
  niter
callbacks:
  cb.evaluation.log()
# of features: 40 
niter: 100
nfeatures : 40 
evaluation_log:
print(xgb.k5.results)
# reshape data for plotting
xgb.k5.results_long <- melt(xgb.k5.results, id.vars = 'Fold')

# plot for each metric
p1 <- plot_metric(xgb.k5.results_long, 'MSE')
p2 <- plot_metric(xgb.k5.results_long, 'RMSE')
p3 <- plot_metric(xgb.k5.results_long, 'MAE')
p4 <- plot_metric(xgb.k5.results_long, 'R2')

# arrange the plots in a 2x2 grid
grid.arrange(p1, p2, p3, p4, ncol = 2, nrow = 2)


plot(p1)

plot(p2)

plot(p3)

plot(p4)

K fold using K=10:

# define the number of folds for cross-validation
num_folds <- 10
folds <- createFolds(train_data$int_rate, k = num_folds, list = TRUE)

K fold using K=10 and linear regression:

#### Linear Regresion applying Cross Validation with k=10  ####

# initialize lists to store models and their results
lm.k10.models <- list()
lm.k10.results <- data.frame()

# perform k-fold cross-validation
for(i in seq_along(folds)) {
  # split the data into training and testing for the current fold
  train_indices <- folds[[i]]
  test_indices <- setdiff(seq_len(nrow(train_data)), train_indices)
  
  train_data_fold <- train_data[train_indices, ]
  test_data_fold <- train_data[test_indices, ]
  
  # fit the model on the training fold
  lm.k10 <- lm(int_rate ~ ., data = train_data_fold)
  lm.k10.models[[i]] <- lm.k10  # Store the model
  
  # make predictions on the training and testing fold
  lm.k10.train_predictions <- predict(lm.k10, newdata = train_data_fold)
  lm.k10.test_predictions <- predict(lm.k10, newdata = test_data_fold)
  
  # calculate metrics for training fold
  lm.k10.train_mse <- mean((lm.k10.train_predictions - train_data_fold$int_rate)^2)
  lm.k10.train_rmse <- sqrt(lm.k10.train_mse)
  lm.k10.train_mae <- mean(abs(lm.k10.train_predictions - train_data_fold$int_rate))
  lm.k10.train_r2 <- summary(lm.k10)$r.squared
  
  # calculate metrics for testing fold
  lm.k10.test_mse <- mean((lm.k10.test_predictions - test_data_fold$int_rate)^2)
  lm.k10.test_rmse <- sqrt(lm.k10.test_mse)
  lm.k10.test_mae <- mean(abs(lm.k10.test_predictions - test_data_fold$int_rate))
  lm.k10.test_r2 <- 1 - (sum((test_data_fold$int_rate - lm.k10.test_predictions)^2) / sum((test_data_fold$int_rate - mean(test_data_fold$int_rate))^2))
  
  # store metrics in the results dataframe
  lm.k10.results <- rbind(lm.k10.results, data.frame(
    Fold = i,
    Train_MSE = lm.k10.train_mse, Test_MSE = lm.k10.test_mse,
    Train_RMSE = lm.k10.train_rmse, Test_RMSE = lm.k10.test_rmse,
    Train_MAE = lm.k10.train_mae, Test_MAE = lm.k10.test_mae,
    Train_R2 = lm.k10.train_r2, Test_R2 = lm.k10.test_r2
  ))
}

# display the models and their metrics
print(lm.k10.models)
[[1]]

Call:
lm(formula = int_rate ~ ., data = train_data_fold)

Coefficients:
                    (Intercept)                        loan_amnt                             term  
                      1.837e+00                        2.761e-05                        3.914e+00  
                     emp_length                   home_ownership              verification_status  
                      1.407e-02                        2.499e-01                        7.342e-01  
                        purpose                       addr_state                      delinq_2yrs  
                      3.385e-01                        1.386e-04                        3.439e-02  
               earliest_cr_line                   inq_last_6mths                         open_acc  
                      1.875e-09                        9.893e-01                        6.491e-02  
                        pub_rec                        revol_bal                       revol_util  
                      4.515e-01                        8.460e-06                        4.138e-02  
                      total_acc              initial_list_status       collections_12_mths_ex_med  
                     -3.711e-02                       -1.047e+00                        2.295e-01  
                 acc_now_delinq                     tot_coll_amt                      tot_cur_bal  
                      1.391e+00                        3.117e-05                       -1.433e-06  
                    open_acc_6m                       open_il_6m                      open_il_12m  
                     -1.691e-02                       -1.541e-01                        1.003e+00  
                    open_il_24m                     total_bal_il                          il_util  
                     -1.838e-01                        2.623e-06                        1.102e-02  
                    open_rv_12m                      open_rv_24m                       max_bal_bc  
                      2.018e-01                        4.167e-02                        1.694e-06  
                       all_util                 total_rev_hi_lim                           inq_fi  
                     -8.827e-03                       -1.778e-05                        7.364e-02  
                    total_cu_tl                     inq_last_12m                annual_inc_merged  
                     -2.619e-02                        9.273e-02                       -2.354e-06  
                     dti_merged            mths_since_delinq_cat       mths_since_last_record_cat  
                      5.175e-02                       -2.058e-01                       -1.284e-01  
         mths_since_rcnt_il_cat  mths_since_last_major_derog_cat  
                      3.155e-01                       -1.363e-01  


[[2]]

Call:
lm(formula = int_rate ~ ., data = train_data_fold)

Coefficients:
                    (Intercept)                        loan_amnt                             term  
                      2.392e+00                        3.706e-05                        3.841e+00  
                     emp_length                   home_ownership              verification_status  
                      1.703e-02                        2.717e-01                        7.160e-01  
                        purpose                       addr_state                      delinq_2yrs  
                      3.370e-01                       -5.944e-05                        3.843e-02  
               earliest_cr_line                   inq_last_6mths                         open_acc  
                      1.881e-09                        1.000e+00                        6.417e-02  
                        pub_rec                        revol_bal                       revol_util  
                      3.828e-01                        6.538e-06                        4.193e-02  
                      total_acc              initial_list_status       collections_12_mths_ex_med  
                     -3.469e-02                       -9.695e-01                        2.127e-01  
                 acc_now_delinq                     tot_coll_amt                      tot_cur_bal  
                      1.530e+00                        1.858e-05                       -6.860e-07  
                    open_acc_6m                       open_il_6m                      open_il_12m  
                      5.075e-02                       -1.453e-01                        1.046e+00  
                    open_il_24m                     total_bal_il                          il_util  
                     -1.060e-01                        1.655e-06                        5.956e-03  
                    open_rv_12m                      open_rv_24m                       max_bal_bc  
                      2.677e-01                        5.487e-02                       -5.448e-05  
                       all_util                 total_rev_hi_lim                           inq_fi  
                     -5.830e-03                       -1.835e-05                        1.787e-01  
                    total_cu_tl                     inq_last_12m                annual_inc_merged  
                     -5.512e-02                        4.549e-02                       -6.257e-06  
                     dti_merged            mths_since_delinq_cat       mths_since_last_record_cat  
                      4.889e-02                       -1.998e-01                       -1.771e-01  
         mths_since_rcnt_il_cat  mths_since_last_major_derog_cat  
                      2.853e-01                       -1.501e-01  


[[3]]

Call:
lm(formula = int_rate ~ ., data = train_data_fold)

Coefficients:
                    (Intercept)                        loan_amnt                             term  
                      2.022e+00                        3.483e-05                        3.833e+00  
                     emp_length                   home_ownership              verification_status  
                      1.891e-02                        2.816e-01                        7.334e-01  
                        purpose                       addr_state                      delinq_2yrs  
                      3.325e-01                        3.149e-04                        5.353e-02  
               earliest_cr_line                   inq_last_6mths                         open_acc  
                      1.881e-09                        9.790e-01                        6.751e-02  
                        pub_rec                        revol_bal                       revol_util  
                      3.643e-01                        5.100e-06                        4.308e-02  
                      total_acc              initial_list_status       collections_12_mths_ex_med  
                     -3.489e-02                       -9.994e-01                        9.544e-02  
                 acc_now_delinq                     tot_coll_amt                      tot_cur_bal  
                      7.195e-01                        3.676e-05                       -1.399e-06  
                    open_acc_6m                       open_il_6m                      open_il_12m  
                      8.692e-02                       -1.618e-01                        5.872e-01  
                    open_il_24m                     total_bal_il                          il_util  
                      8.364e-02                        2.175e-07                        3.842e-03  
                    open_rv_12m                      open_rv_24m                       max_bal_bc  
                      8.637e-02                        7.590e-02                       -2.945e-05  
                       all_util                 total_rev_hi_lim                           inq_fi  
                     -1.163e-04                       -1.710e-05                       -4.120e-02  
                    total_cu_tl                     inq_last_12m                annual_inc_merged  
                     -5.019e-02                        8.224e-02                       -2.299e-06  
                     dti_merged            mths_since_delinq_cat       mths_since_last_record_cat  
                      5.562e-02                       -1.904e-01                       -1.829e-01  
         mths_since_rcnt_il_cat  mths_since_last_major_derog_cat  
                      2.757e-01                       -1.433e-01  


[[4]]

Call:
lm(formula = int_rate ~ ., data = train_data_fold)

Coefficients:
                    (Intercept)                        loan_amnt                             term  
                      1.504e+00                        3.187e-05                        3.908e+00  
                     emp_length                   home_ownership              verification_status  
                      1.735e-02                        2.586e-01                        7.342e-01  
                        purpose                       addr_state                      delinq_2yrs  
                      3.435e-01                        1.434e-03                        3.207e-03  
               earliest_cr_line                   inq_last_6mths                         open_acc  
                      1.850e-09                        9.701e-01                        6.578e-02  
                        pub_rec                        revol_bal                       revol_util  
                      4.056e-01                        3.981e-06                        4.469e-02  
                      total_acc              initial_list_status       collections_12_mths_ex_med  
                     -3.554e-02                       -9.507e-01                        3.245e-01  
                 acc_now_delinq                     tot_coll_amt                      tot_cur_bal  
                      1.578e+00                        2.964e-05                       -1.206e-06  
                    open_acc_6m                       open_il_6m                      open_il_12m  
                     -4.212e-02                       -1.617e-01                        8.188e-01  
                    open_il_24m                     total_bal_il                          il_util  
                      1.138e-01                        4.928e-06                        6.491e-03  
                    open_rv_12m                      open_rv_24m                       max_bal_bc  
                      9.391e-02                        7.342e-02                       -3.985e-05  
                       all_util                 total_rev_hi_lim                           inq_fi  
                     -3.838e-03                       -1.592e-05                       -1.052e-01  
                    total_cu_tl                     inq_last_12m                annual_inc_merged  
                     -2.561e-02                        1.874e-01                       -3.866e-06  
                     dti_merged            mths_since_delinq_cat       mths_since_last_record_cat  
                      4.879e-02                       -1.971e-01                       -1.447e-01  
         mths_since_rcnt_il_cat  mths_since_last_major_derog_cat  
                      3.533e-01                       -1.591e-01  


[[5]]

Call:
lm(formula = int_rate ~ ., data = train_data_fold)

Coefficients:
                    (Intercept)                        loan_amnt                             term  
                      2.209e+00                        3.012e-05                        3.880e+00  
                     emp_length                   home_ownership              verification_status  
                      1.219e-02                        2.479e-01                        7.448e-01  
                        purpose                       addr_state                      delinq_2yrs  
                      3.274e-01                        2.441e-04                        4.065e-02  
               earliest_cr_line                   inq_last_6mths                         open_acc  
                      1.873e-09                        1.008e+00                        5.760e-02  
                        pub_rec                        revol_bal                       revol_util  
                      3.269e-01                        1.939e-06                        4.323e-02  
                      total_acc              initial_list_status       collections_12_mths_ex_med  
                     -3.393e-02                       -1.016e+00                        3.320e-01  
                 acc_now_delinq                     tot_coll_amt                      tot_cur_bal  
                      1.187e+00                        2.533e-05                       -1.187e-06  
                    open_acc_6m                       open_il_6m                      open_il_12m  
                     -5.177e-02                       -1.412e-01                        8.744e-01  
                    open_il_24m                     total_bal_il                          il_util  
                      5.048e-02                        5.570e-08                        2.402e-03  
                    open_rv_12m                      open_rv_24m                       max_bal_bc  
                      1.861e-01                        1.138e-01                       -6.366e-05  
                       all_util                 total_rev_hi_lim                           inq_fi  
                      1.230e-03                       -1.563e-05                        8.822e-02  
                    total_cu_tl                     inq_last_12m                annual_inc_merged  
                     -7.063e-02                        5.952e-02                       -3.328e-06  
                     dti_merged            mths_since_delinq_cat       mths_since_last_record_cat  
                      5.296e-02                       -2.057e-01                       -1.924e-01  
         mths_since_rcnt_il_cat  mths_since_last_major_derog_cat  
                      2.913e-01                       -1.167e-01  


[[6]]

Call:
lm(formula = int_rate ~ ., data = train_data_fold)

Coefficients:
                    (Intercept)                        loan_amnt                             term  
                      1.168e+00                        3.915e-05                        3.871e+00  
                     emp_length                   home_ownership              verification_status  
                      1.352e-02                        2.633e-01                        7.751e-01  
                        purpose                       addr_state                      delinq_2yrs  
                      3.390e-01                        1.749e-03                        1.910e-02  
               earliest_cr_line                   inq_last_6mths                         open_acc  
                      1.832e-09                        9.636e-01                        6.742e-02  
                        pub_rec                        revol_bal                       revol_util  
                      4.285e-01                        3.395e-06                        4.243e-02  
                      total_acc              initial_list_status       collections_12_mths_ex_med  
                     -3.525e-02                       -9.923e-01                        5.245e-01  
                 acc_now_delinq                     tot_coll_amt                      tot_cur_bal  
                      1.433e+00                        3.226e-05                       -7.911e-07  
                    open_acc_6m                       open_il_6m                      open_il_12m  
                     -8.548e-02                       -1.390e-01                        7.381e-01  
                    open_il_24m                     total_bal_il                          il_util  
                      1.076e-01                        2.062e-06                        5.865e-03  
                    open_rv_12m                      open_rv_24m                       max_bal_bc  
                      2.432e-01                        7.391e-02                       -5.362e-05  
                       all_util                 total_rev_hi_lim                           inq_fi  
                      3.292e-03                       -1.637e-05                       -3.937e-03  
                    total_cu_tl                     inq_last_12m                annual_inc_merged  
                     -3.718e-02                        9.189e-02                       -6.204e-06  
                     dti_merged            mths_since_delinq_cat       mths_since_last_record_cat  
                      4.815e-02                       -2.050e-01                       -1.280e-01  
         mths_since_rcnt_il_cat  mths_since_last_major_derog_cat  
                      4.212e-01                       -1.461e-01  


[[7]]

Call:
lm(formula = int_rate ~ ., data = train_data_fold)

Coefficients:
                    (Intercept)                        loan_amnt                             term  
                      2.285e+00                        2.186e-05                        3.935e+00  
                     emp_length                   home_ownership              verification_status  
                      7.481e-03                        2.543e-01                        7.479e-01  
                        purpose                       addr_state                      delinq_2yrs  
                      3.419e-01                       -1.227e-03                        4.564e-02  
               earliest_cr_line                   inq_last_6mths                         open_acc  
                      2.064e-09                        9.941e-01                        4.423e-02  
                        pub_rec                        revol_bal                       revol_util  
                      4.754e-01                       -4.976e-06                        4.734e-02  
                      total_acc              initial_list_status       collections_12_mths_ex_med  
                     -3.345e-02                       -1.095e+00                        2.513e-01  
                 acc_now_delinq                     tot_coll_amt                      tot_cur_bal  
                      1.114e+00                        1.688e-05                       -1.262e-06  
                    open_acc_6m                       open_il_6m                      open_il_12m  
                      1.465e-01                       -1.145e-01                        4.180e-01  
                    open_il_24m                     total_bal_il                          il_util  
                      1.079e-01                        3.507e-06                        1.597e-03  
                    open_rv_12m                      open_rv_24m                       max_bal_bc  
                      1.729e-01                        5.808e-02                       -6.954e-05  
                       all_util                 total_rev_hi_lim                           inq_fi  
                      2.577e-04                       -8.575e-07                        2.366e-02  
                    total_cu_tl                     inq_last_12m                annual_inc_merged  
                     -2.773e-02                        3.311e-02                       -4.980e-06  
                     dti_merged            mths_since_delinq_cat       mths_since_last_record_cat  
                      4.976e-02                       -2.165e-01                       -1.862e-01  
         mths_since_rcnt_il_cat  mths_since_last_major_derog_cat  
                      2.814e-01                       -1.584e-01  


[[8]]

Call:
lm(formula = int_rate ~ ., data = train_data_fold)

Coefficients:
                    (Intercept)                        loan_amnt                             term  
                      1.886e+00                        2.951e-05                        3.939e+00  
                     emp_length                   home_ownership              verification_status  
                      1.907e-02                        2.413e-01                        7.281e-01  
                        purpose                       addr_state                      delinq_2yrs  
                      3.358e-01                       -3.930e-04                        3.378e-02  
               earliest_cr_line                   inq_last_6mths                         open_acc  
                      1.835e-09                        9.790e-01                        6.102e-02  
                        pub_rec                        revol_bal                       revol_util  
                      5.061e-01                        4.566e-06                        4.316e-02  
                      total_acc              initial_list_status       collections_12_mths_ex_med  
                     -3.194e-02                       -1.004e+00                        4.002e-01  
                 acc_now_delinq                     tot_coll_amt                      tot_cur_bal  
                      1.729e+00                        3.088e-05                       -1.232e-06  
                    open_acc_6m                       open_il_6m                      open_il_12m  
                      8.316e-02                       -1.991e-01                        7.761e-01  
                    open_il_24m                     total_bal_il                          il_util  
                      3.994e-02                        3.296e-06                        5.505e-03  
                    open_rv_12m                      open_rv_24m                       max_bal_bc  
                      2.273e-01                        1.900e-05                       -6.886e-05  
                       all_util                 total_rev_hi_lim                           inq_fi  
                      1.625e-03                       -1.555e-05                        1.082e-01  
                    total_cu_tl                     inq_last_12m                annual_inc_merged  
                     -1.459e-01                        6.960e-02                       -3.340e-06  
                     dti_merged            mths_since_delinq_cat       mths_since_last_record_cat  
                      5.213e-02                       -1.958e-01                       -1.031e-01  
         mths_since_rcnt_il_cat  mths_since_last_major_derog_cat  
                      2.445e-01                       -1.429e-01  


[[9]]

Call:
lm(formula = int_rate ~ ., data = train_data_fold)

Coefficients:
                    (Intercept)                        loan_amnt                             term  
                      2.159e+00                        3.265e-05                        3.967e+00  
                     emp_length                   home_ownership              verification_status  
                      1.195e-02                        2.972e-01                        7.602e-01  
                        purpose                       addr_state                      delinq_2yrs  
                      3.471e-01                        6.885e-04                        4.951e-02  
               earliest_cr_line                   inq_last_6mths                         open_acc  
                      1.980e-09                        1.003e+00                        4.107e-02  
                        pub_rec                        revol_bal                       revol_util  
                      4.402e-01                       -3.742e-06                        4.724e-02  
                      total_acc              initial_list_status       collections_12_mths_ex_med  
                     -2.991e-02                       -1.160e+00                        2.061e-01  
                 acc_now_delinq                     tot_coll_amt                      tot_cur_bal  
                      1.381e+00                        2.093e-05                       -8.329e-07  
                    open_acc_6m                       open_il_6m                      open_il_12m  
                      1.517e-01                       -1.813e-01                        4.031e-01  
                    open_il_24m                     total_bal_il                          il_util  
                      1.897e-01                        2.073e-06                        9.850e-03  
                    open_rv_12m                      open_rv_24m                       max_bal_bc  
                      8.659e-02                        5.072e-02                       -1.021e-04  
                       all_util                 total_rev_hi_lim                           inq_fi  
                      1.806e-03                       -1.832e-06                       -2.228e-02  
                    total_cu_tl                     inq_last_12m                annual_inc_merged  
                     -4.495e-02                        1.021e-01                       -8.067e-06  
                     dti_merged            mths_since_delinq_cat       mths_since_last_record_cat  
                      4.395e-02                       -1.989e-01                       -2.025e-01  
         mths_since_rcnt_il_cat  mths_since_last_major_derog_cat  
                      3.087e-01                       -1.601e-01  


[[10]]

Call:
lm(formula = int_rate ~ ., data = train_data_fold)

Coefficients:
                    (Intercept)                        loan_amnt                             term  
                      1.888e+00                        3.751e-05                        3.863e+00  
                     emp_length                   home_ownership              verification_status  
                      1.638e-02                        2.690e-01                        7.461e-01  
                        purpose                       addr_state                      delinq_2yrs  
                      3.473e-01                       -5.930e-04                        4.231e-02  
               earliest_cr_line                   inq_last_6mths                         open_acc  
                      1.968e-09                        9.909e-01                        6.522e-02  
                        pub_rec                        revol_bal                       revol_util  
                      2.888e-01                        4.557e-06                        4.309e-02  
                      total_acc              initial_list_status       collections_12_mths_ex_med  
                     -3.584e-02                       -1.048e+00                        4.556e-01  
                 acc_now_delinq                     tot_coll_amt                      tot_cur_bal  
                      1.046e+00                        2.601e-05                       -1.067e-06  
                    open_acc_6m                       open_il_6m                      open_il_12m  
                     -6.287e-02                       -1.336e-01                        7.843e-01  
                    open_il_24m                     total_bal_il                          il_util  
                      6.702e-02                       -3.403e-06                        6.013e-03  
                    open_rv_12m                      open_rv_24m                       max_bal_bc  
                      3.101e-01                        4.605e-02                       -6.596e-05  
                       all_util                 total_rev_hi_lim                           inq_fi  
                      2.231e-03                       -1.673e-05                        1.676e-01  
                    total_cu_tl                     inq_last_12m                annual_inc_merged  
                     -4.114e-02                        1.939e-02                       -3.840e-06  
                     dti_merged            mths_since_delinq_cat       mths_since_last_record_cat  
                      5.378e-02                       -1.957e-01                       -2.012e-01  
         mths_since_rcnt_il_cat  mths_since_last_major_derog_cat  
                      3.257e-01                       -1.401e-01  
print(lm.k10.results)
plot_metric <- function(results_long, metric) {
    # adjust the variable names based on the metric
    variables <- if (metric == "OOB") {
        "OOB_Error"
    } else {
        c(paste0('Train_', metric), paste0('Test_', metric))
    }
    title <- if (metric == "OOB") {
        paste0(metric, ' per Fold')
    } else {
        paste0('Train vs Test ', metric, ' per Fold')
    }
    
    ggplot(results_long[results_long$variable %in% variables, ],
           aes(x = Fold, y = value, color = variable)) +
    geom_line() +
    geom_point() +
    theme_minimal() +
    labs(title = title,
         x = 'Fold',
         y = metric)
}
# reshape data for plotting
lm.k10.results_long <- melt(lm.k10.results, id.vars = 'Fold')

# plot for each metric
p1 <- plot_metric(lm.k10.results_long, 'MSE')
p2 <- plot_metric(lm.k10.results_long, 'RMSE')
p3 <- plot_metric(lm.k10.results_long, 'MAE')
p4 <- plot_metric(lm.k10.results_long, 'R2')

# arrange the plots in a 2x2 grid
grid.arrange(p1, p2, p3, p4, ncol = 2, nrow = 2)


plot(p1)

plot(p2)

plot(p3)

plot(p4)

K fold using K=10 and Random Forest:

# #### Random Forest applying Cross Validation with k=10  ####
# 
# # initialize lists to store models and their results
# rf.k10.models <- list()
# rf.k10.results <- data.frame()
# 
# # perform k-fold cross-validation
# for(i in seq_along(folds)) {
#   # split the data into training and testing for the current fold
#   train_indices <- folds[[i]]
#   test_indices <- setdiff(seq_len(nrow(train_data)), train_indices)
#   
#   train_data_fold <- train_data[train_indices, ]
#   test_data_fold <- train_data[test_indices, ]
#   
#   # fit the model on the training fold
#   rf.k10 <- ranger(formula = int_rate ~ ., data = train_data, num.trees = 500, verbose=TRUE, importance = "impurity", oob.error = TRUE)
#   rf.k10.models[[i]] <- rf.k10  # Store the model
#   
#   # make predictions on the training and testing fold
#   rf.k10.train_predictions <- predict(rf.k10, data = train_data_fold)$predictions
#   rf.k10.test_predictions <- predict(rf.k10, data = test_data_fold)$predictions
#   
#   # calculate metrics for training fold
#   rf.k10.train_mse <- mean((rf.k10.train_predictions - train_data_fold$int_rate)^2)
#   rf.k10.train_rmse <- sqrt(rf.k10.train_mse)
#   rf.k10.train_mae <- mean(abs(rf.k10.train_predictions - train_data_fold$int_rate))
#   rf.k10.oob_error <- rf.k10$prediction.error
#   
#   # calculate metrics for testing fold
#   rf.k10.test_mse <- mean((rf.k10.test_predictions - test_data_fold$int_rate)^2)
#   rf.k10.test_rmse <- sqrt(rf.k10.test_mse)
#   rf.k10.test_mae <- mean(abs(rf.k10.test_predictions - test_data_fold$int_rate))
#   rf.k10.test_r2 <- 1 - (sum((test_data_fold$int_rate - rf.k10.test_predictions)^2) / sum((test_data_fold$int_rate - mean(test_data_fold$int_rate))^2))
#   
#   # store metrics in the results dataframe
#   rf.k10.results <- rbind(rf.k10.results, data.frame(
#     Fold = i,
#     Train_MSE = rf.k10.train_mse, Test_MSE = rf.k10.test_mse,
#     Train_RMSE = rf.k10.train_rmse, Test_RMSE = rf.k10.test_rmse,
#     Train_MAE = rf.k10.train_mae, Test_MAE = rf.k10.test_mae,
#     OOB_Error = rf.k10.oob_error
#   ))
# }
# 
# # display the models and their metrics
# print(rf.k10.models)
# print(rf.k10.results)
# reshape data for plotting
# rf.k10.results_long <- melt(rf.k10.results, id.vars = 'Fold')
# 
# # plot for each metric
# p1 <- plot_metric(rf.k10.results_long, 'MSE')
# p2 <- plot_metric(rf.k10.results_long, 'RMSE')
# p3 <- plot_metric(rf.k10.results_long, 'MAE')
# p4 <- plot_metric(rf.k10.results_long, 'OOB')
# 
# # arrange the plots in a 2x2 grid
# grid.arrange(p1, p2, p3, p4, ncol = 2, nrow = 2)
# 
# plot(p1)
# plot(p2)
# plot(p3)
# plot(p4)

K fold using K=10 and Boosting:

#### Boosting applying Cross Validation with k=10  ####

# initialize lists to store models and their results
xgb.k10.models <- list()
xgb.k10.results <- data.frame()

# perform k-fold cross-validation
for(i in seq_along(folds)) {
  # split the data into training and testing for the current fold
  train_indices <- folds[[i]]
  test_indices <- setdiff(seq_len(nrow(train_data)), train_indices)
  
  train_data_fold <- train_data[train_indices, ]
  test_data_fold <- train_data[test_indices, ]
  
  # prepare data for xgboost
  xgb.y_train_fold <- train_data_fold$int_rate
  xgb.X_train_fold <- as.matrix(train_data_fold[, -which(names(train_data_fold) == 'int_rate')])
  
  xgb.y_test_fold <- test_data_fold$int_rate
  xgb.X_test_fold <- as.matrix(test_data_fold[, -which(names(test_data_fold) == 'int_rate')])
  
  # fit the xgboost model on the training fold
  xgb.k10 <- xgboost(
    data = xgb.X_train_fold,
    label = xgb.y_train_fold,
    nrounds = 100,
    verbose = 0
  )
  xgb.k10.models[[i]] <- xgb.k10  # store the model
  
  # make predictions on the training fold
  xgb.k10.train_predictions <- predict(xgb.k10, newdata = xgb.X_train_fold)
  # make predictions on the testing fold
  xgb.k10.test_predictions <- predict(xgb.k10, newdata = xgb.X_test_fold)
  
  # calculate metrics for training fold
  xgb.k10.train_mse <- mean((xgb.k10.train_predictions - train_data_fold$int_rate)^2)
  xgb.k10.train_rmse <- sqrt(xgb.k10.train_mse)
  xgb.k10.train_mae <- mean(abs(xgb.k10.train_predictions - train_data_fold$int_rate))
  xgb.k10.train_r2 <- 1 - (sum((xgb.y_train_fold - xgb.k10.train_predictions)^2) / sum((xgb.y_train_fold - mean(xgb.y_train_fold))^2))

  # calculate metrics for testing fold
  xgb.k10.test_mse <- mean((xgb.k10.test_predictions - xgb.y_test_fold)^2)
  xgb.k10.test_rmse <- sqrt(xgb.k10.test_mse)
  xgb.k10.test_mae <- mean(abs(xgb.k10.test_predictions - xgb.y_test_fold))
  xgb.k10.test_r2 <- 1 - (sum((xgb.y_test_fold - xgb.k10.test_predictions)^2) / sum((xgb.y_test_fold - mean(xgb.y_test_fold))^2))  
  
  # store metrics in the results dataframe
  xgb.k10.results <- rbind(xgb.k10.results, data.frame(
    Fold = i,
    Train_MSE = xgb.k10.train_mse, Test_MSE = xgb.k10.test_mse,
    Train_RMSE = xgb.k10.train_rmse, Test_RMSE = xgb.k10.test_rmse,
    Train_MAE = xgb.k10.train_mae, Test_MAE = xgb.k10.test_mae,
    Train_R2 = xgb.k10.train_r2, Test_R2 = xgb.k10.test_r2
  ))
}

# display the models and their metrics
print(xgb.k10.models)
[[1]]
##### xgb.Booster
raw: 432.1 Kb 
call:
  xgb.train(params = params, data = dtrain, nrounds = nrounds, 
    watchlist = watchlist, verbose = verbose, print_every_n = print_every_n, 
    early_stopping_rounds = early_stopping_rounds, maximize = maximize, 
    save_period = save_period, save_name = save_name, xgb_model = xgb_model, 
    callbacks = callbacks)
params (as set within xgb.train):
  validate_parameters = "TRUE"
xgb.attributes:
  niter
callbacks:
  cb.evaluation.log()
# of features: 40 
niter: 100
nfeatures : 40 
evaluation_log:

[[2]]
##### xgb.Booster
raw: 432.4 Kb 
call:
  xgb.train(params = params, data = dtrain, nrounds = nrounds, 
    watchlist = watchlist, verbose = verbose, print_every_n = print_every_n, 
    early_stopping_rounds = early_stopping_rounds, maximize = maximize, 
    save_period = save_period, save_name = save_name, xgb_model = xgb_model, 
    callbacks = callbacks)
params (as set within xgb.train):
  validate_parameters = "TRUE"
xgb.attributes:
  niter
callbacks:
  cb.evaluation.log()
# of features: 40 
niter: 100
nfeatures : 40 
evaluation_log:

[[3]]
##### xgb.Booster
raw: 433.4 Kb 
call:
  xgb.train(params = params, data = dtrain, nrounds = nrounds, 
    watchlist = watchlist, verbose = verbose, print_every_n = print_every_n, 
    early_stopping_rounds = early_stopping_rounds, maximize = maximize, 
    save_period = save_period, save_name = save_name, xgb_model = xgb_model, 
    callbacks = callbacks)
params (as set within xgb.train):
  validate_parameters = "TRUE"
xgb.attributes:
  niter
callbacks:
  cb.evaluation.log()
# of features: 40 
niter: 100
nfeatures : 40 
evaluation_log:

[[4]]
##### xgb.Booster
raw: 432.1 Kb 
call:
  xgb.train(params = params, data = dtrain, nrounds = nrounds, 
    watchlist = watchlist, verbose = verbose, print_every_n = print_every_n, 
    early_stopping_rounds = early_stopping_rounds, maximize = maximize, 
    save_period = save_period, save_name = save_name, xgb_model = xgb_model, 
    callbacks = callbacks)
params (as set within xgb.train):
  validate_parameters = "TRUE"
xgb.attributes:
  niter
callbacks:
  cb.evaluation.log()
# of features: 40 
niter: 100
nfeatures : 40 
evaluation_log:

[[5]]
##### xgb.Booster
raw: 438.5 Kb 
call:
  xgb.train(params = params, data = dtrain, nrounds = nrounds, 
    watchlist = watchlist, verbose = verbose, print_every_n = print_every_n, 
    early_stopping_rounds = early_stopping_rounds, maximize = maximize, 
    save_period = save_period, save_name = save_name, xgb_model = xgb_model, 
    callbacks = callbacks)
params (as set within xgb.train):
  validate_parameters = "TRUE"
xgb.attributes:
  niter
callbacks:
  cb.evaluation.log()
# of features: 40 
niter: 100
nfeatures : 40 
evaluation_log:

[[6]]
##### xgb.Booster
raw: 433.8 Kb 
call:
  xgb.train(params = params, data = dtrain, nrounds = nrounds, 
    watchlist = watchlist, verbose = verbose, print_every_n = print_every_n, 
    early_stopping_rounds = early_stopping_rounds, maximize = maximize, 
    save_period = save_period, save_name = save_name, xgb_model = xgb_model, 
    callbacks = callbacks)
params (as set within xgb.train):
  validate_parameters = "TRUE"
xgb.attributes:
  niter
callbacks:
  cb.evaluation.log()
# of features: 40 
niter: 100
nfeatures : 40 
evaluation_log:

[[7]]
##### xgb.Booster
raw: 438.8 Kb 
call:
  xgb.train(params = params, data = dtrain, nrounds = nrounds, 
    watchlist = watchlist, verbose = verbose, print_every_n = print_every_n, 
    early_stopping_rounds = early_stopping_rounds, maximize = maximize, 
    save_period = save_period, save_name = save_name, xgb_model = xgb_model, 
    callbacks = callbacks)
params (as set within xgb.train):
  validate_parameters = "TRUE"
xgb.attributes:
  niter
callbacks:
  cb.evaluation.log()
# of features: 40 
niter: 100
nfeatures : 40 
evaluation_log:

[[8]]
##### xgb.Booster
raw: 438.9 Kb 
call:
  xgb.train(params = params, data = dtrain, nrounds = nrounds, 
    watchlist = watchlist, verbose = verbose, print_every_n = print_every_n, 
    early_stopping_rounds = early_stopping_rounds, maximize = maximize, 
    save_period = save_period, save_name = save_name, xgb_model = xgb_model, 
    callbacks = callbacks)
params (as set within xgb.train):
  validate_parameters = "TRUE"
xgb.attributes:
  niter
callbacks:
  cb.evaluation.log()
# of features: 40 
niter: 100
nfeatures : 40 
evaluation_log:

[[9]]
##### xgb.Booster
raw: 441.5 Kb 
call:
  xgb.train(params = params, data = dtrain, nrounds = nrounds, 
    watchlist = watchlist, verbose = verbose, print_every_n = print_every_n, 
    early_stopping_rounds = early_stopping_rounds, maximize = maximize, 
    save_period = save_period, save_name = save_name, xgb_model = xgb_model, 
    callbacks = callbacks)
params (as set within xgb.train):
  validate_parameters = "TRUE"
xgb.attributes:
  niter
callbacks:
  cb.evaluation.log()
# of features: 40 
niter: 100
nfeatures : 40 
evaluation_log:

[[10]]
##### xgb.Booster
raw: 437.6 Kb 
call:
  xgb.train(params = params, data = dtrain, nrounds = nrounds, 
    watchlist = watchlist, verbose = verbose, print_every_n = print_every_n, 
    early_stopping_rounds = early_stopping_rounds, maximize = maximize, 
    save_period = save_period, save_name = save_name, xgb_model = xgb_model, 
    callbacks = callbacks)
params (as set within xgb.train):
  validate_parameters = "TRUE"
xgb.attributes:
  niter
callbacks:
  cb.evaluation.log()
# of features: 40 
niter: 100
nfeatures : 40 
evaluation_log:
print(xgb.k10.results)
# reshape data for plotting
xgb.k10.results_long <- melt(xgb.k10.results, id.vars = 'Fold')

# plot for each metric
p1 <- plot_metric(xgb.k10.results_long, 'MSE')
p2 <- plot_metric(xgb.k10.results_long, 'RMSE')
p3 <- plot_metric(xgb.k10.results_long, 'MAE')
p4 <- plot_metric(xgb.k10.results_long, 'R2')

# arrange the plots in a 2x2 grid
grid.arrange(p1, p2, p3, p4, ncol = 2, nrow = 2)


plot(p1)

plot(p2)

plot(p3)

plot(p4)

Decision Trees

#### Decision Trees ####

# error in tree: "factor predictors must have at most 32 levels" is thrown

# basically, it becomes computationally expensive to create so many splits in your data, since you are selecting the best split out of all 2^32 (approx) possible splits

# fit a decision tree model on the training data
#tm <- tree(int_rate ~ ., data = train_data)

# make predictions on the training and testing data
#tm.train_predictions <- predict(tm, newdata = train_data)
#tm.test_predictions <- predict(tm, newdata = test_data)

# calculate Mean Squared Error (MSE) for training and testing
#tm.train_mse <- mean((tm.train_predictions - train_data$int_rate)^2)
#tm.test_mse <- mean((tm.test_predictions - test_data$int_rate)^2)

# calculate Root Mean Squared Error (RMSE) for training and testing
#tm.train_rmse <- sqrt(tm.train_mse)
#tm.test_rmse <- sqrt(tm.test_mse)

# calculate Mean Absolute Error (MAE) for training and testing
#tm.train_mae <- mean(abs(tm.train_predictions - train_data$int_rate))
#tm.test_mae <- mean(abs(tm.test_predictions - test_data$int_rate))

# calculate R-squared (R²) for training and testing
#tm.train_r2 <- 1 - (sum((train_data$int_rate - tm.train_predictions)^2) / sum((train_data$int_rate - mean(train_data$int_rate))^2))
#tm.test_r2 <- 1 - (sum((test_data$int_rate - tm.test_predictions)^2) / sum((test_data$int_rate - mean(test_data$int_rate))^2))

# display the metrics
#cat("Training MSE:", tm.train_mse, "\n")
#cat("Testing MSE:", tm.test_mse, "\n")
#cat("Training RMSE:", tm.train_rmse, "\n")
#cat("Testing RMSE:", tm.test_rmse, "\n")
#cat("Training MAE:", tm.train_mae, "\n")
#cat("Testing MAE:", tm.test_mae, "\n")
#cat("Training R-squared (R²):", tm.train_r2, "\n")
#cat("Testing R-squared (R²):", tm.test_r2, "\n")

Random Forest

#### Random Forest ####

# train a Random Forest model
rf <- ranger(formula = int_rate ~ ., data = train_data, num.trees = 500, verbose=TRUE, importance = "impurity", oob.error = TRUE)
Growing trees.. Progress: 16%. Estimated remaining time: 2 minutes, 42 seconds.
Growing trees.. Progress: 32%. Estimated remaining time: 2 minutes, 12 seconds.
Growing trees.. Progress: 49%. Estimated remaining time: 1 minute, 38 seconds.
Growing trees.. Progress: 65%. Estimated remaining time: 1 minute, 7 seconds.
Growing trees.. Progress: 81%. Estimated remaining time: 35 seconds.
Growing trees.. Progress: 98%. Estimated remaining time: 4 seconds.
# print the model summary
print("Random Forest Model Summary:")
[1] "Random Forest Model Summary:"
print(rf)
Ranger result

Call:
 ranger(formula = int_rate ~ ., data = train_data, num.trees = 500,      verbose = TRUE, importance = "impurity", oob.error = TRUE) 

Type:                             Regression 
Number of trees:                  500 
Sample size:                      638392 
Number of independent variables:  40 
Mtry:                             6 
Target node size:                 5 
Variable importance mode:         impurity 
Splitrule:                        variance 
OOB prediction error (MSE):       8.747259 
R squared (OOB):                  0.5445284 
# make predictions on the training and testing data
rf.train_predictions <- predict(rf, data = train_data)
rf.test_predictions <- predict(rf, data = test_data)

# calculate Mean Squared Error (MSE) for training and testing
rf.train_mse <- mean((rf.train_predictions$predictions - train_data$int_rate)^2)
rf.test_mse <- mean((rf.test_predictions$predictions - test_data$int_rate)^2)

# calculate Root Mean Squared Error (RMSE) for training and testing
rf.train_rmse <- sqrt(rf.train_mse)
rf.test_rmse <- sqrt(rf.test_mse)

# calculate Mean Absolute Error (MAE) for training and testing
rf.train_mae <- mean(abs(rf.train_predictions$predictions - train_data$int_rate))
rf.test_mae <- mean(abs(rf.test_predictions$predictions - test_data$int_rate))

# calculate R-squared (R²) for training and testing
rf.train_r2 <- 1 - (sum((train_data$int_rate - rf.train_predictions$predictions)^2) / sum((train_data$int_rate - mean(train_data$int_rate))^2))
rf.test_r2 <- 1 - (sum((test_data$int_rate - rf.test_predictions$predictions)^2) / sum((test_data$int_rate - mean(test_data$int_rate))^2))

# display the metrics
cat("Training MSE:", rf.train_mse, "\n")
Training MSE: 2.075461 
cat("Testing MSE:", rf.test_mse, "\n")
Testing MSE: 8.69482 
cat("Training RMSE:", rf.train_rmse, "\n")
Training RMSE: 1.440646 
cat("Testing RMSE:", rf.test_rmse, "\n")
Testing RMSE: 2.948698 
cat("Training MAE:", rf.train_mae, "\n")
Training MAE: 1.132674 
cat("Testing MAE:", rf.test_mae, "\n")
Testing MAE: 2.331985 
cat("Training R-squared (R²):", rf.train_r2, "\n")
Training R-squared (R²): 0.8919301 
cat("Testing R-squared (R²):", rf.test_r2, "\n")
Testing R-squared (R²): 0.5470498 

Boosting

#### Boosting ####

# define the target variable for training and testing
xgb.y_train <- train_data$int_rate
xgb.y_test <- test_data$int_rate

# define the feature matrix for training and testing (exclude the target variable)
xgb.X_train <- train_data[, -which(names(train_data) == 'int_rate')]
xgb.X_test <- test_data[, -which(names(test_data) == 'int_rate')]

# fit a gradient boosting regression model using xgboost
xgb <- xgboost(
  data = as.matrix(xgb.X_train),
  label = xgb.y_train,
  nrounds = 100,
  verbose = 0
)

# make predictions on the training and testing data
xgb.train_predictions <- predict(xgb, newdata = as.matrix(xgb.X_train))
xgb.test_predictions <- predict(xgb, newdata = as.matrix(xgb.X_test))

# calculate Mean Squared Error (MSE) for training and testing
xgb.train_mse <- mean((xgb.train_predictions - xgb.y_train)^2)
xgb.test_mse <- mean((xgb.test_predictions - xgb.y_test)^2)

# calculate Root Mean Squared Error (RMSE) for training and testing
xgb.train_rmse <- sqrt(xgb.train_mse)
xgb.test_rmse <- sqrt(xgb.test_mse)

# calculate Mean Absolute Error (MAE) for training and testing
xgb.train_mae <- mean(abs(xgb.train_predictions - xgb.y_train))
xgb.test_mae <- mean(abs(xgb.test_predictions - xgb.y_test))

# calculate R-squared (R²) for training and testing
xgb.train_r2 <- 1 - (sum((xgb.y_train - xgb.train_predictions)^2) / sum((xgb.y_train - mean(xgb.y_train))^2))
xgb.test_r2 <- 1 - (sum((xgb.y_test - xgb.test_predictions)^2) / sum((xgb.y_test - mean(xgb.y_test))^2))

# display the metrics
cat("Training MSE:", xgb.train_mse, "\n")
Training MSE: 7.708131 
cat("Testing MSE:", xgb.test_mse, "\n")
Testing MSE: 8.015791 
cat("Training RMSE:", xgb.train_rmse, "\n")
Training RMSE: 2.776352 
cat("Testing RMSE:", xgb.test_rmse, "\n")
Testing RMSE: 2.831217 
cat("Training MAE:", xgb.train_mae, "\n")
Training MAE: 2.178155 
cat("Testing MAE:", xgb.test_mae, "\n")
Testing MAE: 2.220999 
cat("Training R-squared (R²):", xgb.train_r2, "\n")
Training R-squared (R²): 0.5986354 
cat("Testing R-squared (R²):", xgb.test_r2, "\n")
Testing R-squared (R²): 0.5824233 

Following, a scatter plot of actual vs predicted training values for each model is plot. This plot helps us visualize how well each model’s predictions align with the actual data points.

# create a scatter plot function
create_scatter_plot <- function(actual_values, predicted_values, model_name) {
  model_comparison_data <- data.frame(
    Actual = actual_values,
    Predicted = predicted_values
  )
  
  scatter_plot <- ggplot(model_comparison_data, aes(x = Actual, y = Predicted)) +
    geom_point() +
    geom_abline(intercept = 0, slope = 1, linetype = "dashed", color = "red") +  # add a diagonal reference line
    labs(x = "Actual Training Values", y = "Predicted Training Values", title = model_name) +
    theme_minimal() +
    ylim(-50, 50)
  
  return(scatter_plot)
}

# create scatter plots for each model
lm_scatter_plot <- create_scatter_plot(
  actual_values = train_data$int_rate,
  predicted_values = lm.train_predictions,
  model_name = "Linear Regression"
)

rf_scatter_plot <- create_scatter_plot(
  actual_values = train_data$int_rate,
  predicted_values = rf.train_predictions$predictions,
  model_name = "Random Forest"
)

xgb_scatter_plot <- create_scatter_plot(
  actual_values = xgb.y_train,
  predicted_values = xgb.train_predictions,
  model_name = "XGBoost"
)

# display the scatter plots separately
print(lm_scatter_plot)

print(rf_scatter_plot)

print(xgb_scatter_plot)

Following, a scatter plot of actual vs predicted testing values for each model is plot. This plot helps us visualize how well each model’s predictions align with the actual data points.

# create a scatter plot function
create_scatter_plot <- function(actual_values, predicted_values, model_name) {
  model_comparison_data <- data.frame(
    Actual = actual_values,
    Predicted = predicted_values
  )
  
  scatter_plot <- ggplot(model_comparison_data, aes(x = Actual, y = Predicted)) +
    geom_point() +
    geom_abline(intercept = 0, slope = 1, linetype = "dashed", color = "red") +  # add a diagonal reference line
    labs(x = "Actual Testing Values", y = "Predicted Testing Values", title = model_name) +
    theme_minimal() +
    ylim(-50, 50) +
    xlim(0, 40)
  
  return(scatter_plot)
}

# create scatter plots for each model
lm_scatter_plot <- create_scatter_plot(
  actual_values = test_data$int_rate,
  predicted_values = lm.test_predictions,
  model_name = "Linear Regression"
)

rf_scatter_plot <- create_scatter_plot(
  actual_values = test_data$int_rate,
  predicted_values = rf.test_predictions$predictions,
  model_name = "Random Forest"
)

xgb_scatter_plot <- create_scatter_plot(
  actual_values = xgb.y_test,
  predicted_values = xgb.test_predictions,
  model_name = "XGBoost"
)

# display the scatter plots separately
print(lm_scatter_plot)

print(rf_scatter_plot)

print(xgb_scatter_plot)

Residual plots can help identify patterns in prediction errors and assess whether the assumptions of linear regression (if applicable) are met.

# create a residual plot function
create_residual_plot <- function(actual_values, predicted_values, model_name) {
  residuals <- actual_values - predicted_values
  residual_data <- data.frame(
    Predicted = predicted_values,
    Residuals = residuals
  )
  
  residual_plot <- ggplot(residual_data, aes(x = Predicted, y = Residuals)) +
    geom_point() +
    geom_hline(yintercept = 0, linetype = "dashed", color = "red") +  # Red horizontal reference line
    labs(x = "Predicted Values", y = "Residuals", title = paste("Residual Plot -", model_name)) +
    theme_minimal() +
    ylim(-30, 30) +
    xlim(0, 40)
  
  return(residual_plot)
}

# create residual plots for each model
lm_residual_plot <- create_residual_plot(
  actual_values = train_data$int_rate,
  predicted_values = lm.train_predictions,
  model_name = "Linear Regression"
)

rf_residual_plot <- create_residual_plot(
  actual_values = train_data$int_rate,
  predicted_values = rf.train_predictions$predictions,
  model_name = "Random Forest"
)

xgb_residual_plot <- create_residual_plot(
  actual_values = xgb.y_train,
  predicted_values = xgb.train_predictions,
  model_name = "XGBoost"
)

# display the residual plots separately
print(lm_residual_plot)

print(rf_residual_plot)

print(xgb_residual_plot)

# create a density plot function for residuals
create_residual_density_plot <- function(actual_values, predicted_values, model_name) {
  residuals <- actual_values - predicted_values
  residual_data <- data.frame(Residuals = residuals)
  
  density_plot <- ggplot(residual_data, aes(x = Residuals)) +
    geom_density(fill = "skyblue", color = "black", alpha = 0.7) +
    labs(x = "Residuals", y = "Density", title = paste("Residual Density Plot -", model_name)) +
    theme_minimal() +
    xlim(-30,30) + 
    ylim(0, 0.35)
    
  
  return(density_plot)
}

# create density plots for residuals for each model
lm_residual_density_plot <- create_residual_density_plot(
  actual_values = train_data$int_rate,
  predicted_values = lm.train_predictions,
  model_name = "Linear Regression"
)

rf_residual_density_plot <- create_residual_density_plot(
  actual_values = train_data$int_rate,
  predicted_values = rf.train_predictions$predictions,
  model_name = "Random Forest"
)

xgb_residual_density_plot <- create_residual_density_plot(
  actual_values = xgb.y_train,
  predicted_values = xgb.train_predictions,
  model_name = "XGBoost"
)

# display the density plots separately
print(lm_residual_density_plot)

print(rf_residual_density_plot)

print(xgb_residual_density_plot)

This visualization can help you compare the distribution of prediction errors across models through histograms.

# create a histogram plot function for residuals with a red density curve
create_residual_histogram_plot <- function(actual_values, predicted_values, model_name) {
  residuals <- actual_values - predicted_values
  residual_data <- data.frame(Residuals = residuals)
  
  histogram_plot <- ggplot(residual_data, aes(x = Residuals)) +
    geom_histogram(aes(y = after_stat(density)), bins = 30, fill = "skyblue", color = "black", alpha = 0.7) +  # use density on the y-axis for the histogram
    geom_density(color = "red", linewidth = 1.5) +  # add the density plot in red
    labs(x = "Residuals", y = "Density", title = paste("Residual Histogram Plot with Density Curve -", model_name)) +
    theme_minimal() +
    xlim(-20,20) + 
    ylim(0, 0.3)
  
  return(histogram_plot)
}

# create histogram plots for residuals for each model
lm_residual_histogram_plot <- create_residual_histogram_plot(
  actual_values = train_data$int_rate,
  predicted_values = lm.train_predictions,
  model_name = "Linear Regression"
)

rf_residual_histogram_plot <- create_residual_histogram_plot(
  actual_values = train_data$int_rate,
  predicted_values = rf.train_predictions$predictions,
  model_name = "Random Forest"
)

xgb_residual_histogram_plot <- create_residual_histogram_plot(
  actual_values = xgb.y_train,
  predicted_values = xgb.train_predictions,
  model_name = "XGBoost"
)

# display the histogram plots separately
print(lm_residual_histogram_plot)

print(rf_residual_histogram_plot)

print(xgb_residual_histogram_plot)

For each model a bar chart that displays the R-squared (coefficient of determination) values is created. R-squared measures the proportion of variance in the target variable explained by the model. Higher R-squared values indicate better model fit.

# create a data frame with R-squared values for each model
model_names <- c("Linear Regression", "Random Forest", "XGBoost")
r_squared_values_train <- c(
  lm.train_r2,
  rf.train_r2,
  xgb.train_r2
)
r_squared_values_test <- c(
  lm.test_r2,
  rf.test_r2,
  xgb.test_r2
)

r_squared_data_train <- data.frame(Model = factor(model_names),
                              R_squared = r_squared_values_train)
r_squared_data_test <- data.frame(Model = factor(model_names),
                              R_squared = r_squared_values_test)

# create the R-squared comparison bar chart
r_squared_bar_chart_train <- ggplot(r_squared_data_train, aes(x = Model, y = R_squared, fill = Model)) +
  geom_bar(stat = "identity") +
  labs(x = "Model", y = "R-squared (R²)", title = "R-squared Comparison Training") +
  theme_minimal() +
  theme(axis.text.x = element_text(angle = 45, hjust = 1)) + 
  ylim(0,1)
r_squared_bar_chart_test <- ggplot(r_squared_data_test, aes(x = Model, y = R_squared, fill = Model)) +
  geom_bar(stat = "identity") +
  labs(x = "Model", y = "R-squared (R²)", title = "R-squared Comparison Testing") +
  theme_minimal() +
  theme(axis.text.x = element_text(angle = 45, hjust = 1)) + 
  ylim(0,1)

# display the R-squared comparison bar chart
print(r_squared_bar_chart_train)

print(r_squared_bar_chart_test)

A bar chart that compares the MAE or RMSE values, is generated for each model. These metrics quantify the average prediction errors of each model, and lower values are preferred.

# create a data frame with MAE and RMSE values for each model
model_names <- c("Linear Regression", "Random Forest", "XGBoost","Linear Regression", "Random Forest", "XGBoost")
error_values_train <- c(
  lm.train_mae,
  rf.train_mae,
  xgb.train_mae,
  lm.train_rmse,
  rf.train_rmse,
  xgb.train_rmse
)
error_values_test <- c(
  lm.test_mae,
  rf.test_mae,
  xgb.test_mae,
  lm.test_rmse,
  rf.test_rmse,
  xgb.test_rmse
)
error_type <- c(
  "MAE", "MAE", "MAE","RMSE","RMSE","RMSE"
)
model_errors_train <- data.frame(Model = factor(model_names, levels = c("Linear Regression", "Random Forest", "XGBoost")),
                Error = error_values_train, Type = error_type)
model_errors_test <- data.frame(Model = factor(model_names, levels = c("Linear Regression", "Random Forest", "XGBoost")),
                Error = error_values_test, Type = error_type)
# create the MAE or RMSE comparison bar chart
error_bar_chart_train <- ggplot(model_errors_train, aes(x = Model, y = Error, fill = Type)) +
  geom_bar(stat = "identity", position = "dodge") +
  labs(x = "Model", y = "Error Value", title = "Training MAE and RMSE Comparison") +
  theme_minimal() +
  theme(axis.text.x = element_text(angle = 45, hjust = 1)) + 
  ylim(0, 4)

error_bar_chart_test <- ggplot(model_errors_test, aes(x = Model, y = Error, fill = Type)) +
  geom_bar(stat = "identity", position = "dodge") +
  labs(x = "Model", y = "Error Value", title = "Testing MAE and RMSE Comparison") +
  theme_minimal() +
  theme(axis.text.x = element_text(angle = 45, hjust = 1)) + 
  ylim(0, 4)

# display the MAE and RMSE comparison bar chart
print(error_bar_chart_train)

print(error_bar_chart_test)

#### Random Forest Feature Importance Plot ####
v1 <- vip(rf, title = "Ranger", num_features = 20) 
plot(v1)

Feature Selection from the variable importance’s analysis:

imp.variables <- lc_data[, v1$data$Variable]
imp.variables$int_rate <- lc_data$int_rate
imp.train_indices <- createDataPartition(imp.variables$int_rate, p = 0.8, list = FALSE)

# create training and testing datasets
imp.train_data <- imp.variables[imp.train_indices, ]
imp.test_data <- imp.variables[-imp.train_indices, ]
#### Linear Regression with only importance variables ####

imp.lm.fit <- lm(int_rate ~ ., data = imp.train_data)

# make predictions on the training and testing data
imp.lm.train_predictions <- predict(imp.lm.fit, newdata = imp.train_data)
imp.lm.test_predictions <- predict(imp.lm.fit, newdata = imp.test_data)

# calculate Mean Squared Error (MSE) for training and testing
imp.lm.train_mse <- mean((imp.lm.train_predictions - imp.train_data$int_rate)^2)
imp.lm.test_mse <- mean((imp.lm.test_predictions - imp.test_data$int_rate)^2)

# calculate Root Mean Squared Error (RMSE) for training and testing
imp.lm.train_rmse <- sqrt(imp.lm.train_mse)
imp.lm.test_rmse <- sqrt(imp.lm.test_mse)

# calculate Mean Absolute Error (MAE) for training and testing
imp.lm.train_mae <- mean(abs(imp.lm.train_predictions - imp.train_data$int_rate))
imp.lm.test_mae <- mean(abs(imp.lm.test_predictions - imp.test_data$int_rate))

# calculate R-squared (R²) for training and testing
imp.lm.train_r2 <- 1 - (sum((imp.train_data$int_rate - imp.lm.train_predictions)^2) / sum((imp.train_data$int_rate - mean(imp.train_data$int_rate))^2))
imp.lm.test_r2 <- 1 - (sum((imp.test_data$int_rate - imp.lm.test_predictions)^2) / sum((imp.test_data$int_rate - mean(imp.test_data$int_rate))^2))

# display the metrics
cat("Training MSE:", imp.lm.train_mse, "\n")
Training MSE: 10.86566 
cat("Testing MSE:", imp.lm.test_mse, "\n")
Testing MSE: 10.99868 
cat("Training RMSE:", imp.lm.train_rmse, "\n")
Training RMSE: 3.29631 
cat("Testing RMSE:", imp.lm.test_rmse, "\n")
Testing RMSE: 3.316426 
cat("Training MAE:", imp.lm.train_mae, "\n")
Training MAE: 2.61514 
cat("Testing MAE:", imp.lm.test_mae, "\n")
Testing MAE: 2.613664 
cat("Training R-squared (R²):", imp.lm.train_r2, "\n")
Training R-squared (R²): 0.4346627 
cat("Testing R-squared (R²):", imp.lm.test_r2, "\n")
Testing R-squared (R²): 0.425238 
#### Random Forest with only importance variables ####

# train a Random Forest model
imp.rf <- ranger(formula = int_rate ~ ., data = imp.train_data, num.trees = 500, verbose=TRUE, importance = "impurity", oob.error = TRUE)
Growing trees.. Progress: 21%. Estimated remaining time: 1 minute, 59 seconds.
Growing trees.. Progress: 42%. Estimated remaining time: 1 minute, 26 seconds.
Growing trees.. Progress: 64%. Estimated remaining time: 53 seconds.
Growing trees.. Progress: 85%. Estimated remaining time: 22 seconds.
# print the model summary
print("Random Forest Model Summary:")
[1] "Random Forest Model Summary:"
print(imp.rf)
Ranger result

Call:
 ranger(formula = int_rate ~ ., data = imp.train_data, num.trees = 500,      verbose = TRUE, importance = "impurity", oob.error = TRUE) 

Type:                             Regression 
Number of trees:                  500 
Sample size:                      638392 
Number of independent variables:  20 
Mtry:                             4 
Target node size:                 5 
Variable importance mode:         impurity 
Splitrule:                        variance 
OOB prediction error (MSE):       8.743306 
R squared (OOB):                  0.5450889 
# make predictions on the training and testing data
imp.rf.train_predictions <- predict(imp.rf, data = imp.train_data)
imp.rf.test_predictions <- predict(imp.rf, data = imp.test_data)

# calculate Mean Squared Error (MSE) for training and testing
imp.rf.train_mse <- mean((imp.rf.train_predictions$predictions - imp.train_data$int_rate)^2)
imp.rf.test_mse <- mean((imp.rf.test_predictions$predictions - imp.test_data$int_rate)^2)

# calculate Root Mean Squared Error (RMSE) for training and testing
imp.rf.train_rmse <- sqrt(imp.rf.train_mse)
imp.rf.test_rmse <- sqrt(imp.rf.test_mse)

# calculate Mean Absolute Error (MAE) for training and testing
imp.rf.train_mae <- mean(abs(imp.rf.train_predictions$predictions - imp.train_data$int_rate))
imp.rf.test_mae <- mean(abs(imp.rf.test_predictions$predictions - imp.test_data$int_rate))

# calculate R-squared (R²) for training and testing
imp.rf.train_r2 <- 1 - (sum((imp.train_data$int_rate - imp.rf.train_predictions$predictions)^2) / sum((imp.train_data$int_rate - mean(imp.train_data$int_rate))^2))
imp.rf.test_r2 <- 1 - (sum((test_data$int_rate - rf.test_predictions$predictions)^2) / sum((imp.test_data$int_rate - mean(imp.test_data$int_rate))^2))

# display the metrics
cat("Training MSE:", imp.rf.train_mse, "\n")
Training MSE: 1.714178 
cat("Testing MSE:", imp.rf.test_mse, "\n")
Testing MSE: 8.674378 
cat("Training RMSE:", imp.rf.train_rmse, "\n")
Training RMSE: 1.309266 
cat("Testing RMSE:", imp.rf.test_rmse, "\n")
Testing RMSE: 2.94523 
cat("Training MAE:", imp.rf.train_mae, "\n")
Training MAE: 1.02214 
cat("Testing MAE:", imp.rf.test_mae, "\n")
Testing MAE: 2.329954 
cat("Training R-squared (R²):", imp.rf.train_r2, "\n")
Training R-squared (R²): 0.9108118 
cat("Testing R-squared (R²):", imp.rf.test_r2, "\n")
Testing R-squared (R²): 0.5456317 
#### Boosting with only importance variables ####

# define the target variable for training and testing
imp.xgb.y_train <- imp.train_data$int_rate
imp.xgb.y_test <- imp.test_data$int_rate

# define the feature matrix for training and testing (exclude the target variable)
imp.xgb.X_train <- imp.train_data[, -which(names(imp.train_data) == 'int_rate')]
imp.xgb.X_test <- imp.test_data[, -which(names(imp.test_data) == 'int_rate')]

# fit a gradient boosting regression model using xgboost
imp.xgb <- xgboost(
  data = as.matrix(imp.xgb.X_train),
  label = imp.xgb.y_train,
  nrounds = 100,
  verbose = 0
)

# make predictions on the training and testing data
imp.xgb.train_predictions <- predict(imp.xgb, newdata = as.matrix(imp.xgb.X_train))
imp.xgb.test_predictions <- predict(imp.xgb, newdata = as.matrix(imp.xgb.X_test))

# calculate Mean Squared Error (MSE) for training and testing
imp.xgb.train_mse <- mean((imp.xgb.train_predictions - imp.xgb.y_train)^2)
imp.xgb.test_mse <- mean((imp.xgb.test_predictions - imp.xgb.y_test)^2)

# calculate Root Mean Squared Error (RMSE) for training and testing
imp.xgb.train_rmse <- sqrt(imp.xgb.train_mse)
imp.xgb.test_rmse <- sqrt(imp.xgb.test_mse)

# calculate Mean Absolute Error (MAE) for training and testing
imp.xgb.train_mae <- mean(abs(imp.xgb.train_predictions - imp.xgb.y_train))
imp.xgb.test_mae <- mean(abs(imp.xgb.test_predictions - imp.xgb.y_test))

# calculate R-squared (R²) for training and testing
imp.xgb.train_r2 <- 1 - (sum((imp.xgb.y_train - imp.xgb.train_predictions)^2) / sum((imp.xgb.y_train - mean(imp.xgb.y_train))^2))
imp.xgb.test_r2 <- 1 - (sum((imp.xgb.y_test - imp.xgb.test_predictions)^2) / sum((imp.xgb.y_test - mean(imp.xgb.y_test))^2))

# display the metrics
cat("Training MSE:", imp.xgb.train_mse, "\n")
Training MSE: 7.75411 
cat("Testing MSE:", imp.xgb.test_mse, "\n")
Testing MSE: 8.068693 
cat("Training RMSE:", imp.xgb.train_rmse, "\n")
Training RMSE: 2.78462 
cat("Testing RMSE:", imp.xgb.test_rmse, "\n")
Testing RMSE: 2.840544 
cat("Training MAE:", imp.xgb.train_mae, "\n")
Training MAE: 2.186026 
cat("Testing MAE:", imp.xgb.test_mae, "\n")
Testing MAE: 2.231548 
cat("Training R-squared (R²):", imp.xgb.train_r2, "\n")
Training R-squared (R²): 0.5965558 
cat("Testing R-squared (R²):", imp.xgb.test_r2, "\n")
Testing R-squared (R²): 0.5783514 

The dataset was filtered by the 20 variables with the most importance (from the rf results). As we can see above, the errors of each model are more or less the errors with the double variables we had before, so filtering by these 20 “important variables” does not seem making sense…

Hyperparameter Tuning for XGBoosting:

---
title: "R Notebook"
output: html_notebook
---

# Data Pre-processing

Load needed libraries

```{r}
library(readr)
library(ggplot2)
library(dplyr)
library(caret)
library(glmnet)
library(boot)
library(tree)
library(ranger)
library(xgboost)
library(gbm)
library(vip)
library(ISLR)
library(tidyr)
library(gridExtra)
library(reshape2)
```
Set the seed for reproducibility

```{r}
set.seed(1)
```

Load the dataset

```{r}
original_lc_data <- read.csv("LCdata.csv",sep = ";")
lc_data <- original_lc_data
```

Remove attributes not available for prediction

```{r}
lc_data <- subset(lc_data, select = -c(collection_recovery_fee, installment, issue_d,
                                       last_pymnt_amnt, last_pymnt_d, loan_status,
                                       next_pymnt_d, out_prncp, out_prncp_inv,
                                       pymnt_plan, recoveries, total_pymnt,
                                       total_pymnt_inv,total_rec_int, total_rec_late_fee, 
                                       total_rec_prncp))
```

```{r}
summary(lc_data)
```
First we delete the columns which aren't useful for our prediction

```{r}
lc_data$id <- NULL
lc_data$member_id <- NULL
lc_data$zip_code <- NULL
lc_data$url <- NULL
```

Looks like **policy_code** contains just value equal to 1, it can be removed

```{r}
lc_data$policy_code <- NULL
```

Remove additional columns which are related to the historical data

```{r}
lc_data$last_credit_pull_d <- NULL
```

Then we delete the columns which can't be converted to categorical and require NLP

```{r}
lc_data$title <- NULL
lc_data$desc <- NULL
lc_data$emp_title <- NULL
```

Let's examine the **loan_amnt** column

```{r}
sum(is.na(lc_data$loan_amnt))
cor(lc_data$loan_amnt, lc_data$int_rate)
hist(lc_data$loan_amnt, breaks = 20, main = "loan_amnt distribution", xlab = "loan_amnt", col = "lightblue", border = "black")
ggplot(data = lc_data, mapping = aes(x=int_rate,y=loan_amnt)) + geom_boxplot()
```

Standardize **loan_amnt**

```{r}
#lc_data$loan_amnt <- scale(lc_data$loan_amnt)
```

Let's examine the **funded_amnt** column

```{r}
sum(is.na(lc_data$funded_amnt))
cor(lc_data$funded_amnt, lc_data$int_rate)
hist(lc_data$funded_amnt, breaks = 20, main = "funded_amnt distribution", xlab = "funded_amnt", col = "lightblue", border = "black")
```

As we can see, **funded_amnt** is almost the same as the **loan_amnt** column, consequently, we remove it.

```{r}
lc_data$funded_amnt <- NULL 
```

Let's examine the **funded_amnt_inv** column

```{r}
sum(is.na(lc_data$funded_amnt_inv))
cor(lc_data$funded_amnt_inv, lc_data$int_rate)
hist(lc_data$funded_amnt_inv, breaks = 20, main = "funded_amnt_inv distribution", xlab = "funded_amnt_inv", col = "lightblue", border = "black")
```

Remove **funded_amnt_inv** for the same reason as above

```{r}
lc_data$funded_amnt_inv <- NULL
```

Let's see the **int_rate** distribution.
```{r}
hist(lc_data$int_rate, breaks = 20, main = "int_rate distribution", xlab = "int_rate", col = "lightblue", border = "black")
```

Standardize int rate:
```{r}
#lc_data$int_rate <- scale(lc_data$int_rate)
```

As we can observe, there are 40363 NAs. We can assume 40363 do not work.
```{r}
barplot(table(lc_data$emp_length),
        xlab = "emp_length years", 
        ylab = "Frequency", 
        col = "skyblue", 
        border = "black",
        cex.names = 0.6)  # The size of the main title
```

Since **emp_length** seems to be categorical, we transform it to as a factor and then as numeric.
The conversion to numeric is needed for supporting the XGBoost

```{r}
lc_data$emp_length <- as.factor(lc_data$emp_length)
ggplot(data = lc_data, mapping = aes(x=int_rate,y=emp_length)) + geom_boxplot()
lc_data$emp_length <- as.numeric(lc_data$emp_length)
```

As we can see, **term** plays a crucial role in predicting the interest rate.

```{r}
lc_data$term <- as.factor(lc_data$term)
ggplot(data = lc_data, mapping = aes(x=int_rate,y=term)) + geom_boxplot()
lc_data$term <- as.numeric(lc_data$term)
```

Cleaning of **home_ownership**:

During the data cleaning phase, our analysis revealed that the variable "home_ownership" does not show a distinct correlation with interest rates. Specifically, among the categories, "ANY" and "OTHER" contain 2 and 154 cases, respectively, while the "NONE" category comprises 39 cases. Although the "NONE" category appears to demonstrate a higher interest rate compared to others, the limited sample size of 39 cases raises doubts about the reliability of this observation. Notably, the "NONE" category might pertain to individuals experiencing homelessness, prompting ethical concerns about loan provision to this demographic.

```{r}
table(lc_data$home_ownership)
ggplot(data = lc_data, mapping = aes(x=int_rate,y=home_ownership)) + geom_boxplot()
```

Then, we retain mortgage, own and rent:

```{r}
lc_data <- lc_data %>% filter(home_ownership %in% c("MORTGAGE","OWN","RENT"))
lc_data$home_ownership <- as.numeric(as.factor(lc_data$home_ownership))
```

Application joint handling:

```{r}

# merging annual income
lc_data <- lc_data %>% mutate(
    annual_inc_merged = ifelse(is.na(annual_inc_joint)== TRUE, annual_inc,annual_inc_joint)) 

lc_data <- lc_data %>% select(-annual_inc,-annual_inc_joint)

# merging debt to income ratio
lc_data <- lc_data %>% mutate(
    dti_merged = ifelse(is.na(dti_joint)== TRUE, dti,dti_joint)) 

lc_data <- lc_data %>% select(-dti,-dti_joint)

```

Upon reviewing the summary again, it becomes apparent that there are merely 460 joint applications, constituting a small subset within the extensive dataset of around 800k rows. Through consolidating the debt-to-income ratios (dti's), we can pinpoint the data pertinent to our research objectives. Hence, it is advisable to eliminate the columns verification_status_joint and application_type to prevent introducing unwarranted variability into our analysis.

```{r}
table(lc_data$verification_status)
table(lc_data$verification_status_joint)
```

```{r}
lc_data$verification_status <- as.numeric(as.factor(lc_data$verification_status))
lc_data <- lc_data %>% select(-verification_status_joint, -application_type)
```


Let's check if other is NA or a real value for purpose. It's a real one, so we don't have to handle it.
```{r}
lc_data$purpose <- as.factor(lc_data$purpose)
ggplot(data = lc_data, mapping = aes(x=int_rate,y=purpose)) + geom_boxplot()
lc_data$purpose <- as.numeric(lc_data$purpose)
```

Let's have a glance to the state address:
```{r}
table(lc_data$addr_state)
lc_data$addr_state <- as.factor(lc_data$addr_state)
ggplot(data = lc_data, mapping = aes(x=int_rate,y=addr_state)) + geom_boxplot()
lc_data$addr_state <- as.numeric(lc_data$addr_state)
```

Regarding delinquency in the last 2 years, there are few NAs then remove them:
```{r}
lc_data <- lc_data %>% 
    filter(!(is.na(delinq_2yrs)))
```

The columns **mths_since_delinq_cat**, **mths_since_last_record**, **mths_since_rcnt_il** and **mths_since_last_major_derog** contain numerical values which refer to the number of the months. Since this columns contain a lot of null values which can't be replaced with 0's, one of the most appropriate operations that can be made is applying discretization. We do this by creating a set of contiguous bins based on years, while for the null values we create a separate bin.

```{r}
lc_data <- lc_data %>%
  mutate(mths_since_delinq_cat = ifelse(
    is.na(mths_since_last_delinq) == TRUE,
    "NONE",
    ifelse(
      mths_since_last_delinq <= 12,
      "Less_1_Y",
      ifelse(
        mths_since_last_delinq <= 24,
        "Less_2_Y",
        ifelse(
          mths_since_last_delinq <= 36,
          "Less_3_Y",
          ifelse(mths_since_last_delinq <= 48, "Less_4_Y", "More_4_Y")
        )
      )
    )
  )) %>% select(-mths_since_last_delinq)
          
lc_data$mths_since_delinq_cat <- as.factor(lc_data$mths_since_delinq_cat)
ggplot(data = lc_data, mapping = aes(x=int_rate,y=mths_since_delinq_cat))+geom_boxplot()
lc_data$mths_since_delinq_cat <- as.numeric(lc_data$mths_since_delinq_cat)
```

```{r}
lc_data <- lc_data %>%
  mutate(mths_since_last_record_cat = ifelse(
    is.na(mths_since_last_record) == TRUE,
    "NONE",
    ifelse(
      mths_since_last_record <= 12,
      "Less_1_Y",
      ifelse(
        mths_since_last_record <= 24,
        "Less_2_Y",
        ifelse(
          mths_since_last_record <= 36,
          "Less_3_Y",
          ifelse(mths_since_last_record <= 48, "Less_4_Y", "More_4_Y")
        )
      )
    )
  )) %>% select(-mths_since_last_record)

lc_data$mths_since_last_record_cat <- as.factor(lc_data$mths_since_last_record_cat)
ggplot(data = lc_data, mapping = aes(x=int_rate,y=mths_since_last_record_cat))+geom_boxplot()
lc_data$mths_since_last_record_cat <- as.numeric(lc_data$mths_since_last_record_cat)
```

```{r}
lc_data <-lc_data %>% 
  mutate(mths_since_rcnt_il_cat =  ifelse(
    is.na(mths_since_rcnt_il) == TRUE,
    "NONE",
    ifelse(
      mths_since_rcnt_il <= 12,
      "Less_1_Y",
      ifelse(
        mths_since_rcnt_il <= 24,
        "Less_2_Y",
        ifelse(
          mths_since_rcnt_il <= 36,
          "Less_3_Y",
          ifelse(mths_since_rcnt_il <= 48, "Less_4_Y", "More_4_Y")
        )
      )
    )
  )) %>% select(-mths_since_rcnt_il)

lc_data$mths_since_rcnt_il_cat <- as.factor(lc_data$mths_since_rcnt_il_cat)
ggplot(data = lc_data, mapping = aes(x=int_rate,y=mths_since_rcnt_il_cat))+geom_boxplot()
lc_data$mths_since_rcnt_il_cat <- as.numeric(lc_data$mths_since_rcnt_il_cat)
```

```{r}
lc_data <-lc_data %>% 
  mutate(mths_since_last_major_derog_cat =  ifelse(
    is.na(mths_since_last_major_derog) == TRUE,
    "NONE",
    ifelse(
      mths_since_last_major_derog <= 12,
      "Less_1_Y",
      ifelse(
        mths_since_last_major_derog <= 24,
        "Less_2_Y",
        ifelse(
          mths_since_last_major_derog <= 36,
          "Less_3_Y",
          ifelse(mths_since_last_major_derog <= 48, "Less_4_Y", "More_4_Y")
        )
      )
    )
  )) %>% select(-mths_since_last_major_derog)

lc_data$mths_since_last_major_derog_cat <- as.factor(lc_data$mths_since_last_major_derog_cat)
ggplot(data = lc_data, mapping = aes(x=int_rate,y=mths_since_last_major_derog_cat))+geom_boxplot()
lc_data$mths_since_last_major_derog_cat <- as.numeric(lc_data$mths_since_last_major_derog_cat)

```


The variable **initial_list_status** identifies whether a loan was initially listed in the whole (W) or fractional (F) market. This variable could be useful so we can keep it and transform it to a factor and then to a numeric value, for the same purpose of compatibility with the XGBoost function.

```{r}
lc_data$initial_list_status <- as.factor(lc_data$initial_list_status)
ggplot(data = lc_data, mapping = aes(x=int_rate,y=initial_list_status))+geom_boxplot()
lc_data$initial_list_status <- as.numeric(lc_data$initial_list_status)
```

Let's check which columns still have null values
```{r}
colSums(is.na(lc_data))
```

The columns **revol_bal** and **revol_util** contain only few NA values, those values can't be replaced with 0, then we filter the values which are not NAs.

```{r}
lc_data <- lc_data %>% 
    filter(!(is.na(revol_bal))) %>% 
        filter(!(is.na(revol_util)))
```


Let's check which columns still have null values:

```{r}
names(which(colSums(is.na(lc_data)) > 0))
```

Replace null values with 0 where is possible

```{r}
lc_data <-
  lc_data %>%
  mutate(open_acc_6m = ifelse(is.na(open_acc_6m) == TRUE, 0, open_acc_6m)) %>%
  mutate(tot_cur_bal = ifelse(is.na(tot_cur_bal) == TRUE, 0, tot_cur_bal)) %>%
  mutate(open_il_6m = ifelse(is.na(open_il_6m) == TRUE, 0, open_il_6m)) %>%
  mutate(open_il_12m = ifelse(is.na(open_il_12m) == TRUE, 0, open_il_12m)) %>%
  mutate(open_il_24m = ifelse(is.na(open_il_24m) == TRUE, 0, open_il_24m)) %>%
  mutate(total_bal_il = ifelse(is.na(total_bal_il) == TRUE, 0, total_bal_il)) %>%
  mutate(il_util = ifelse(is.na(il_util) == TRUE, 0, il_util)) %>%
  mutate(open_rv_12m = ifelse(is.na(open_rv_12m) == TRUE, 0, open_rv_12m)) %>%
  mutate(total_rev_hi_lim = ifelse(is.na(total_rev_hi_lim) == TRUE, 0, total_rev_hi_lim)) %>%
  mutate(max_bal_bc = ifelse(is.na(max_bal_bc) == TRUE, 0, max_bal_bc)) %>%
  mutate(all_util = ifelse(is.na(all_util) == TRUE, 0, all_util)) %>%
  mutate(inq_fi = ifelse(is.na(inq_fi) == TRUE, 0, inq_fi)) %>%
  mutate(total_cu_tl = ifelse(is.na(total_cu_tl) == TRUE, 0, total_cu_tl)) %>%
  mutate(inq_last_12m = ifelse(is.na(inq_last_12m) == TRUE, 0, inq_last_12m)) %>%
  mutate(open_rv_24m = ifelse(is.na(open_rv_24m) == TRUE, 0, open_rv_24m)) %>%
  mutate(tot_coll_amt = ifelse(is.na(tot_coll_amt)== TRUE,0, tot_coll_amt)) %>%
  mutate(collections_12_mths_ex_med = ifelse(is.na(collections_12_mths_ex_med)== TRUE,0, collections_12_mths_ex_med))
```

**earliest_cr_line** contains the month the borrower's earliest reported credit line was opened.
Even if this date consists only on month and year, still there are too many unique values.
We could transform the dates in to a numerical value, by converting them from date into Unix Time.
This unit measures time by the number of seconds that have elapsed since 00:00:00 UTC on 1 January 1970.
Since this column doesn't contain the day number, we take as a reference the first day of the month.

```{r}
lc_data <- lc_data %>% 
    filter(!(is.na(earliest_cr_line)))

# function to replace dates with unix time
to_unix_time <- function(date) {
  tmp <- paste("01", date, sep="-")
  return (as.numeric(as.POSIXct(tmp, format="%d-%b-%Y", tz="UTC")))
}

# map dates to unix time
lc_data$earliest_cr_line <- apply(lc_data, 1, function(row) to_unix_time(row["earliest_cr_line"]))

# standardize them
#lc_data$earliest_cr_line <- scale(lc_data$earliest_cr_line)
```

Outliers Removal:

```{r}
boxplot(lc_data$int_rate)
# Identify outliers using boxplot
outliers <- boxplot(lc_data$int_rate, plot = FALSE)$out
# Remove outliers from the dataset
lc_data_clean <- lc_data[!lc_data$int_rate %in% outliers, ]
```

```{r}
summary(lc_data)
```

**Learning Algorithms**

```{r}
# create indices for splitting (80% train, 20% test)
train_indices <- createDataPartition(lc_data$int_rate, p = 0.8, list = FALSE)

# create training and testing datasets
train_data <- lc_data[train_indices, ]
test_data <- lc_data[-train_indices, ]
```

```{r}
#### Linear Regression ####

lm.fit <- lm(int_rate ~ ., data = train_data)

# make predictions on the training and testing data
lm.train_predictions <- predict(lm.fit, newdata = train_data)
lm.test_predictions <- predict(lm.fit, newdata = test_data)

# calculate Mean Squared Error (MSE) for training and testing
lm.train_mse <- mean((lm.train_predictions - train_data$int_rate)^2)
lm.test_mse <- mean((lm.test_predictions - test_data$int_rate)^2)

# calculate Root Mean Squared Error (RMSE) for training and testing
lm.train_rmse <- sqrt(lm.train_mse)
lm.test_rmse <- sqrt(lm.test_mse)

# calculate Mean Absolute Error (MAE) for training and testing
lm.train_mae <- mean(abs(lm.train_predictions - train_data$int_rate))
lm.test_mae <- mean(abs(lm.test_predictions - test_data$int_rate))

# calculate R-squared (R²) for training and testing
lm.train_r2 <- 1 - (sum((train_data$int_rate - lm.train_predictions)^2) / sum((train_data$int_rate - mean(train_data$int_rate))^2))
lm.test_r2 <- 1 - (sum((test_data$int_rate - lm.test_predictions)^2) / sum((test_data$int_rate - mean(test_data$int_rate))^2))

# display the metrics
cat("Training MSE:", lm.train_mse, "\n")
cat("Testing MSE:", lm.test_mse, "\n")
cat("Training RMSE:", lm.train_rmse, "\n")
cat("Testing RMSE:", lm.test_rmse, "\n")
cat("Training MAE:", lm.train_mae, "\n")
cat("Testing MAE:", lm.test_mae, "\n")
cat("Training R-squared (R²):", lm.train_r2, "\n")
cat("Testing R-squared (R²):", lm.test_r2, "\n")
```
**Lasso**
It standardizes data automatically

```{r}
lasso.predictors_train <- model.matrix(int_rate ~ ., train_data)[,-1]
lasso.target_train <- train_data$int_rate
lasso.predictors_test <- model.matrix(int_rate ~ ., test_data)[,-1]
lasso.target_test <- test_data$int_rate

lasso.fit <- glmnet(lasso.predictors_train, lasso.target_train, alpha = 1)

plot(lasso.fit, label=TRUE)

# make predictions on the training and testing data
lasso.train_predictions <- predict(lasso.fit, newdata = train_data, newx = lasso.predictors_train)
lasso.test_predictions <- predict(lasso.fit, newdata = test_data, newx = lasso.predictors_train)

# calculate Mean Squared Error (MSE) for training and testing
lasso.train_mse <- mean((lasso.train_predictions - train_data$int_rate)^2)
lasso.test_mse <- mean((lasso.test_predictions - test_data$int_rate)^2)

# calculate Root Mean Squared Error (RMSE) for training and testing
lasso.train_rmse <- sqrt(lasso.train_mse)
lasso.test_rmse <- sqrt(lasso.test_mse)

# calculate Mean Absolute Error (MAE) for training and testing
lasso.train_mae <- mean(abs(lasso.train_predictions - train_data$int_rate))
lasso.test_mae <- mean(abs(lasso.test_predictions - test_data$int_rate))

# calculate R-squared (R²) for training and testing
lasso.train_r2 <- 1 - (sum((train_data$int_rate - lasso.train_predictions)^2) / sum((train_data$int_rate - mean(train_data$int_rate))^2))
lasso.test_r2 <- 1 - (sum((test_data$int_rate - lasso.test_predictions)^2) / sum((test_data$int_rate - mean(test_data$int_rate))^2))

# display the metrics
cat("Training MSE:", lasso.train_mse, "\n")
cat("Testing MSE:", lasso.test_mse, "\n")
cat("Training RMSE:", lasso.train_rmse, "\n")
cat("Testing RMSE:", lasso.test_rmse, "\n")
cat("Training MAE:", lasso.train_mae, "\n")
cat("Testing MAE:", lasso.test_mae, "\n")
cat("Training R-squared (R²):", lasso.train_r2, "\n")
cat("Testing R-squared (R²):", lasso.test_r2, "\n")
```

**Ridge**
It standardizes data automatically

```{r}
ridge.predictors_train <- model.matrix(int_rate ~ ., train_data)[,-1]
ridge.target_train <- train_data$int_rate
ridge.predictors_test <- model.matrix(int_rate ~ ., test_data)[,-1]
ridge.target_test <- test_data$int_rate

ridge.fit <- glmnet(ridge.predictors_train, ridge.target_train, alpha = 0)

plot(ridge.fit, label=TRUE, xlab = "L2 Norm")

# make predictions on the training and testing data
ridge.train_predictions <- predict(ridge.fit, newdata = train_data, newx = ridge.predictors_train)
ridge.test_predictions <- predict(ridge.fit, newdata = test_data, newx = ridge.predictors_train)

# calculate Mean Squared Error (MSE) for training and testing
ridge.train_mse <- mean((ridge.train_predictions - train_data$int_rate)^2)
ridge.test_mse <- mean((ridge.test_predictions - test_data$int_rate)^2)

# calculate Root Mean Squared Error (RMSE) for training and testing
ridge.train_rmse <- sqrt(ridge.train_mse)
ridge.test_rmse <- sqrt(ridge.test_mse)

# calculate Mean Absolute Error (MAE) for training and testing
ridge.train_mae <- mean(abs(ridge.train_predictions - train_data$int_rate))
ridge.test_mae <- mean(abs(ridge.test_predictions - test_data$int_rate))

# calculate R-squared (R²) for training and testing
ridge.train_r2 <- 1 - (sum((train_data$int_rate - ridge.train_predictions)^2) / sum((train_data$int_rate - mean(train_data$int_rate))^2))
ridge.test_r2 <- 1 - (sum((test_data$int_rate - ridge.test_predictions)^2) / sum((test_data$int_rate - mean(test_data$int_rate))^2))

# display the metrics
cat("Training MSE:", ridge.train_mse, "\n")
cat("Testing MSE:", ridge.test_mse, "\n")
cat("Training RMSE:", ridge.train_rmse, "\n")
cat("Testing RMSE:", ridge.test_rmse, "\n")
cat("Training MAE:", ridge.train_mae, "\n")
cat("Testing MAE:", ridge.test_mae, "\n")
cat("Training R-squared (R²):", ridge.train_r2, "\n")
cat("Testing R-squared (R²):", ridge.test_r2, "\n")
```

K fold using K=5:
```{r}
# define the number of folds for cross-validation
num_folds <- 5
folds <- createFolds(train_data$int_rate, k = num_folds, list = TRUE)
```


K fold using K=5 and linear regression:
```{r}
#### Linear Regresion applying Cross Validation with k=5  ####

# initialize lists to store models and their results
lm.k5.models <- list()
lm.k5.results <- data.frame()

# perform k-fold cross-validation
for(i in seq_along(folds)) {
  # split the data into training and testing for the current fold
  train_indices <- folds[[i]]
  test_indices <- setdiff(seq_len(nrow(train_data)), train_indices)
  
  train_data_fold <- train_data[train_indices, ]
  test_data_fold <- train_data[test_indices, ]
  
  # fit the model on the training fold
  lm.k5 <- lm(int_rate ~ ., data = train_data_fold)
  lm.k5.models[[i]] <- lm.k5  # Store the model
  
  # make predictions on the training and testing fold
  lm.k5.train_predictions <- predict(lm.k5, newdata = train_data_fold)
  lm.k5.test_predictions <- predict(lm.k5, newdata = test_data_fold)
  
  # calculate metrics for training fold
  lm.k5.train_mse <- mean((lm.k5.train_predictions - train_data_fold$int_rate)^2)
  lm.k5.train_rmse <- sqrt(lm.k5.train_mse)
  lm.k5.train_mae <- mean(abs(lm.k5.train_predictions - train_data_fold$int_rate))
  lm.k5.train_r2 <- summary(lm.k5)$r.squared
  
  # calculate metrics for testing fold
  lm.k5.test_mse <- mean((lm.k5.test_predictions - test_data_fold$int_rate)^2)
  lm.k5.test_rmse <- sqrt(lm.k5.test_mse)
  lm.k5.test_mae <- mean(abs(lm.k5.test_predictions - test_data_fold$int_rate))
  lm.k5.test_r2 <- 1 - (sum((test_data_fold$int_rate - lm.k5.test_predictions)^2) / sum((test_data_fold$int_rate - mean(test_data_fold$int_rate))^2))
  
  # store metrics in the results dataframe
  lm.k5.results <- rbind(lm.k5.results, data.frame(
    Fold = i,
    Train_MSE = lm.k5.train_mse, Test_MSE = lm.k5.test_mse,
    Train_RMSE = lm.k5.train_rmse, Test_RMSE = lm.k5.test_rmse,
    Train_MAE = lm.k5.train_mae, Test_MAE = lm.k5.test_mae,
    Train_R2 = lm.k5.train_r2, Test_R2 = lm.k5.test_r2
  ))
}

# display the models and their metrics
print(lm.k5.models)
print(lm.k5.results)
```

```{r}
plot_metric <- function(results_long, metric) {
    # adjust the variable names based on the metric
    variables <- if (metric == "OOB") {
        "OOB_Error"
    } else {
        c(paste0('Train_', metric), paste0('Test_', metric))
    }
    title <- if (metric == "OOB") {
        paste0(metric, ' per Fold')
    } else {
        paste0('Train vs Test ', metric, ' per Fold')
    }
    
    ggplot(results_long[results_long$variable %in% variables, ],
           aes(x = Fold, y = value, color = variable)) +
    geom_line() +
    geom_point() +
    theme_minimal() +
    labs(title = title,
         x = 'Fold',
         y = metric)
}
```


```{r}
# reshape data for plotting
lm.k5.results_long <- melt(lm.k5.results, id.vars = 'Fold')

# plot for each metric
p1 <- plot_metric(lm.k5.results_long, 'MSE')
p2 <- plot_metric(lm.k5.results_long, 'RMSE')
p3 <- plot_metric(lm.k5.results_long, 'MAE')
p4 <- plot_metric(lm.k5.results_long, 'R2')

# arrange the plots in a 2x2 grid
grid.arrange(p1, p2, p3, p4, ncol = 2, nrow = 2)

plot(p1)
plot(p2)
plot(p3)
plot(p4)
```

K fold using K=5 and Random Forest:
```{r}
#### Random Forest applying Cross Validation with k=5  ####

# initialize lists to store models and their results
rf.k5.models <- list()
rf.k5.results <- data.frame()

# perform k-fold cross-validation
for(i in seq_along(folds)) {
  # split the data into training and testing for the current fold
  train_indices <- folds[[i]]
  test_indices <- setdiff(seq_len(nrow(train_data)), train_indices)
  
  train_data_fold <- train_data[train_indices, ]
  test_data_fold <- train_data[test_indices, ]
  
  # fit the model on the training fold
  rf.k5 <- ranger(formula = int_rate ~ ., data = train_data, num.trees = 500, verbose=TRUE, importance = "impurity", oob.error = TRUE)
  rf.k5.models[[i]] <- rf.k5  # Store the model
  
  # make predictions on the training and testing fold
  rf.k5.train_predictions <- predict(rf.k5, data = train_data_fold)$predictions
  rf.k5.test_predictions <- predict(rf.k5, data = test_data_fold)$predictions
  
  # calculate metrics for training fold
  rf.k5.train_mse <- mean((rf.k5.train_predictions - train_data_fold$int_rate)^2)
  rf.k5.train_rmse <- sqrt(rf.k5.train_mse)
  rf.k5.train_mae <- mean(abs(rf.k5.train_predictions - train_data_fold$int_rate))
  rf.k5.oob_error <- rf.k5$prediction.error
  
  # calculate metrics for testing fold
  rf.k5.test_mse <- mean((rf.k5.test_predictions - test_data_fold$int_rate)^2)
  rf.k5.test_rmse <- sqrt(rf.k5.test_mse)
  rf.k5.test_mae <- mean(abs(rf.k5.test_predictions - test_data_fold$int_rate))
  rf.k5.test_r2 <- 1 - (sum((test_data_fold$int_rate - rf.k5.test_predictions)^2) / sum((test_data_fold$int_rate - mean(test_data_fold$int_rate))^2))
  
  # store metrics in the results dataframe
  rf.k5.results <- rbind(rf.k5.results, data.frame(
    Fold = i,
    Train_MSE = rf.k5.train_mse, Test_MSE = rf.k5.test_mse,
    Train_RMSE = rf.k5.train_rmse, Test_RMSE = rf.k5.test_rmse,
    Train_MAE = rf.k5.train_mae, Test_MAE = rf.k5.test_mae,
    OOB_Error = rf.k5.oob_error
  ))
}

# display the models and their metrics
print(rf.k5.models)
print(rf.k5.results)
```
```{r}
# reshape data for plotting
rf.k5.results_long <- melt(rf.k5.results, id.vars = 'Fold')

# plot for each metric
p1 <- plot_metric(rf.k5.results_long, 'MSE')
p2 <- plot_metric(rf.k5.results_long, 'RMSE')
p3 <- plot_metric(rf.k5.results_long, 'MAE')
p4 <- plot_metric(rf.k5.results_long, 'OOB')

# arrange the plots in a 2x2 grid
grid.arrange(p1, p2, p3, p4, ncol = 2, nrow = 2)

plot(p1)
plot(p2)
plot(p3)
plot(p4)
```

K fold using K=5 and Boosting:
```{r}
#### Boosting applying Cross Validation with k=5  ####

# initialize lists to store models and their results
xgb.k5.models <- list()
xgb.k5.results <- data.frame()

# perform k-fold cross-validation
for(i in seq_along(folds)) {
  # split the data into training and testing for the current fold
  train_indices <- folds[[i]]
  test_indices <- setdiff(seq_len(nrow(train_data)), train_indices)
  
  train_data_fold <- train_data[train_indices, ]
  test_data_fold <- train_data[test_indices, ]
  
  # prepare data for xgboost
  xgb.y_train_fold <- train_data_fold$int_rate
  xgb.X_train_fold <- as.matrix(train_data_fold[, -which(names(train_data_fold) == 'int_rate')])
  
  xgb.y_test_fold <- test_data_fold$int_rate
  xgb.X_test_fold <- as.matrix(test_data_fold[, -which(names(test_data_fold) == 'int_rate')])
  
  # fit the xgboost model on the training fold
  xgb.k5 <- xgboost(
    data = xgb.X_train_fold,
    label = xgb.y_train_fold,
    nrounds = 100,
    verbose = 0
  )
  xgb.k5.models[[i]] <- xgb.k5  # store the model
  
  # make predictions on the training fold
  xgb.k5.train_predictions <- predict(xgb.k5, newdata = xgb.X_train_fold)
  # make predictions on the testing fold
  xgb.k5.test_predictions <- predict(xgb.k5, newdata = xgb.X_test_fold)
  
  # calculate metrics for training fold
  xgb.k5.train_mse <- mean((xgb.k5.train_predictions - train_data_fold$int_rate)^2)
  xgb.k5.train_rmse <- sqrt(xgb.k5.train_mse)
  xgb.k5.train_mae <- mean(abs(xgb.k5.train_predictions - train_data_fold$int_rate))
  xgb.k5.train_r2 <- 1 - (sum((xgb.y_train_fold - xgb.k5.train_predictions)^2) / sum((xgb.y_train_fold - mean(xgb.y_train_fold))^2))

  # calculate metrics for testing fold
  xgb.k5.test_mse <- mean((xgb.k5.test_predictions - xgb.y_test_fold)^2)
  xgb.k5.test_rmse <- sqrt(xgb.k5.test_mse)
  xgb.k5.test_mae <- mean(abs(xgb.k5.test_predictions - xgb.y_test_fold))
  xgb.k5.test_r2 <- 1 - (sum((xgb.y_test_fold - xgb.k5.test_predictions)^2) / sum((xgb.y_test_fold - mean(xgb.y_test_fold))^2))  
  
  # store metrics in the results dataframe
  xgb.k5.results <- rbind(xgb.k5.results, data.frame(
    Fold = i,
    Train_MSE = xgb.k5.train_mse, Test_MSE = xgb.k5.test_mse,
    Train_RMSE = xgb.k5.train_rmse, Test_RMSE = xgb.k5.test_rmse,
    Train_MAE = xgb.k5.train_mae, Test_MAE = xgb.k5.test_mae,
    Train_R2 = xgb.k5.train_r2, Test_R2 = xgb.k5.test_r2
  ))
}

# display the models and their metrics
print(xgb.k5.models)
print(xgb.k5.results)
```
```{r}
# reshape data for plotting
xgb.k5.results_long <- melt(xgb.k5.results, id.vars = 'Fold')

# plot for each metric
p1 <- plot_metric(xgb.k5.results_long, 'MSE')
p2 <- plot_metric(xgb.k5.results_long, 'RMSE')
p3 <- plot_metric(xgb.k5.results_long, 'MAE')
p4 <- plot_metric(xgb.k5.results_long, 'R2')

# arrange the plots in a 2x2 grid
grid.arrange(p1, p2, p3, p4, ncol = 2, nrow = 2)

plot(p1)
plot(p2)
plot(p3)
plot(p4)
```

K fold using K=10:
```{r}
# define the number of folds for cross-validation
num_folds <- 10
folds <- createFolds(train_data$int_rate, k = num_folds, list = TRUE)
```


K fold using K=10 and linear regression:
```{r}
#### Linear Regresion applying Cross Validation with k=10  ####

# initialize lists to store models and their results
lm.k10.models <- list()
lm.k10.results <- data.frame()

# perform k-fold cross-validation
for(i in seq_along(folds)) {
  # split the data into training and testing for the current fold
  train_indices <- folds[[i]]
  test_indices <- setdiff(seq_len(nrow(train_data)), train_indices)
  
  train_data_fold <- train_data[train_indices, ]
  test_data_fold <- train_data[test_indices, ]
  
  # fit the model on the training fold
  lm.k10 <- lm(int_rate ~ ., data = train_data_fold)
  lm.k10.models[[i]] <- lm.k10  # Store the model
  
  # make predictions on the training and testing fold
  lm.k10.train_predictions <- predict(lm.k10, newdata = train_data_fold)
  lm.k10.test_predictions <- predict(lm.k10, newdata = test_data_fold)
  
  # calculate metrics for training fold
  lm.k10.train_mse <- mean((lm.k10.train_predictions - train_data_fold$int_rate)^2)
  lm.k10.train_rmse <- sqrt(lm.k10.train_mse)
  lm.k10.train_mae <- mean(abs(lm.k10.train_predictions - train_data_fold$int_rate))
  lm.k10.train_r2 <- summary(lm.k10)$r.squared
  
  # calculate metrics for testing fold
  lm.k10.test_mse <- mean((lm.k10.test_predictions - test_data_fold$int_rate)^2)
  lm.k10.test_rmse <- sqrt(lm.k10.test_mse)
  lm.k10.test_mae <- mean(abs(lm.k10.test_predictions - test_data_fold$int_rate))
  lm.k10.test_r2 <- 1 - (sum((test_data_fold$int_rate - lm.k10.test_predictions)^2) / sum((test_data_fold$int_rate - mean(test_data_fold$int_rate))^2))
  
  # store metrics in the results dataframe
  lm.k10.results <- rbind(lm.k10.results, data.frame(
    Fold = i,
    Train_MSE = lm.k10.train_mse, Test_MSE = lm.k10.test_mse,
    Train_RMSE = lm.k10.train_rmse, Test_RMSE = lm.k10.test_rmse,
    Train_MAE = lm.k10.train_mae, Test_MAE = lm.k10.test_mae,
    Train_R2 = lm.k10.train_r2, Test_R2 = lm.k10.test_r2
  ))
}

# display the models and their metrics
print(lm.k10.models)
print(lm.k10.results)
```

```{r}
plot_metric <- function(results_long, metric) {
    # adjust the variable names based on the metric
    variables <- if (metric == "OOB") {
        "OOB_Error"
    } else {
        c(paste0('Train_', metric), paste0('Test_', metric))
    }
    title <- if (metric == "OOB") {
        paste0(metric, ' per Fold')
    } else {
        paste0('Train vs Test ', metric, ' per Fold')
    }
    
    ggplot(results_long[results_long$variable %in% variables, ],
           aes(x = Fold, y = value, color = variable)) +
    geom_line() +
    geom_point() +
    theme_minimal() +
    labs(title = title,
         x = 'Fold',
         y = metric)
}
```


```{r}
# reshape data for plotting
lm.k10.results_long <- melt(lm.k10.results, id.vars = 'Fold')

# plot for each metric
p1 <- plot_metric(lm.k10.results_long, 'MSE')
p2 <- plot_metric(lm.k10.results_long, 'RMSE')
p3 <- plot_metric(lm.k10.results_long, 'MAE')
p4 <- plot_metric(lm.k10.results_long, 'R2')

# arrange the plots in a 2x2 grid
grid.arrange(p1, p2, p3, p4, ncol = 2, nrow = 2)

plot(p1)
plot(p2)
plot(p3)
plot(p4)
```

K fold using K=10 and Random Forest:
```{r}
# #### Random Forest applying Cross Validation with k=10  ####
# 
# # initialize lists to store models and their results
# rf.k10.models <- list()
# rf.k10.results <- data.frame()
# 
# # perform k-fold cross-validation
# for(i in seq_along(folds)) {
#   # split the data into training and testing for the current fold
#   train_indices <- folds[[i]]
#   test_indices <- setdiff(seq_len(nrow(train_data)), train_indices)
#   
#   train_data_fold <- train_data[train_indices, ]
#   test_data_fold <- train_data[test_indices, ]
#   
#   # fit the model on the training fold
#   rf.k10 <- ranger(formula = int_rate ~ ., data = train_data, num.trees = 500, verbose=TRUE, importance = "impurity", oob.error = TRUE)
#   rf.k10.models[[i]] <- rf.k10  # Store the model
#   
#   # make predictions on the training and testing fold
#   rf.k10.train_predictions <- predict(rf.k10, data = train_data_fold)$predictions
#   rf.k10.test_predictions <- predict(rf.k10, data = test_data_fold)$predictions
#   
#   # calculate metrics for training fold
#   rf.k10.train_mse <- mean((rf.k10.train_predictions - train_data_fold$int_rate)^2)
#   rf.k10.train_rmse <- sqrt(rf.k10.train_mse)
#   rf.k10.train_mae <- mean(abs(rf.k10.train_predictions - train_data_fold$int_rate))
#   rf.k10.oob_error <- rf.k10$prediction.error
#   
#   # calculate metrics for testing fold
#   rf.k10.test_mse <- mean((rf.k10.test_predictions - test_data_fold$int_rate)^2)
#   rf.k10.test_rmse <- sqrt(rf.k10.test_mse)
#   rf.k10.test_mae <- mean(abs(rf.k10.test_predictions - test_data_fold$int_rate))
#   rf.k10.test_r2 <- 1 - (sum((test_data_fold$int_rate - rf.k10.test_predictions)^2) / sum((test_data_fold$int_rate - mean(test_data_fold$int_rate))^2))
#   
#   # store metrics in the results dataframe
#   rf.k10.results <- rbind(rf.k10.results, data.frame(
#     Fold = i,
#     Train_MSE = rf.k10.train_mse, Test_MSE = rf.k10.test_mse,
#     Train_RMSE = rf.k10.train_rmse, Test_RMSE = rf.k10.test_rmse,
#     Train_MAE = rf.k10.train_mae, Test_MAE = rf.k10.test_mae,
#     OOB_Error = rf.k10.oob_error
#   ))
# }
# 
# # display the models and their metrics
# print(rf.k10.models)
# print(rf.k10.results)
```
```{r}
# reshape data for plotting
# rf.k10.results_long <- melt(rf.k10.results, id.vars = 'Fold')
# 
# # plot for each metric
# p1 <- plot_metric(rf.k10.results_long, 'MSE')
# p2 <- plot_metric(rf.k10.results_long, 'RMSE')
# p3 <- plot_metric(rf.k10.results_long, 'MAE')
# p4 <- plot_metric(rf.k10.results_long, 'OOB')
# 
# # arrange the plots in a 2x2 grid
# grid.arrange(p1, p2, p3, p4, ncol = 2, nrow = 2)
# 
# plot(p1)
# plot(p2)
# plot(p3)
# plot(p4)
```

K fold using K=10 and Boosting:
```{r}
#### Boosting applying Cross Validation with k=10  ####

# initialize lists to store models and their results
xgb.k10.models <- list()
xgb.k10.results <- data.frame()

# perform k-fold cross-validation
for(i in seq_along(folds)) {
  # split the data into training and testing for the current fold
  train_indices <- folds[[i]]
  test_indices <- setdiff(seq_len(nrow(train_data)), train_indices)
  
  train_data_fold <- train_data[train_indices, ]
  test_data_fold <- train_data[test_indices, ]
  
  # prepare data for xgboost
  xgb.y_train_fold <- train_data_fold$int_rate
  xgb.X_train_fold <- as.matrix(train_data_fold[, -which(names(train_data_fold) == 'int_rate')])
  
  xgb.y_test_fold <- test_data_fold$int_rate
  xgb.X_test_fold <- as.matrix(test_data_fold[, -which(names(test_data_fold) == 'int_rate')])
  
  # fit the xgboost model on the training fold
  xgb.k10 <- xgboost(
    data = xgb.X_train_fold,
    label = xgb.y_train_fold,
    nrounds = 100,
    verbose = 0
  )
  xgb.k10.models[[i]] <- xgb.k10  # store the model
  
  # make predictions on the training fold
  xgb.k10.train_predictions <- predict(xgb.k10, newdata = xgb.X_train_fold)
  # make predictions on the testing fold
  xgb.k10.test_predictions <- predict(xgb.k10, newdata = xgb.X_test_fold)
  
  # calculate metrics for training fold
  xgb.k10.train_mse <- mean((xgb.k10.train_predictions - train_data_fold$int_rate)^2)
  xgb.k10.train_rmse <- sqrt(xgb.k10.train_mse)
  xgb.k10.train_mae <- mean(abs(xgb.k10.train_predictions - train_data_fold$int_rate))
  xgb.k10.train_r2 <- 1 - (sum((xgb.y_train_fold - xgb.k10.train_predictions)^2) / sum((xgb.y_train_fold - mean(xgb.y_train_fold))^2))

  # calculate metrics for testing fold
  xgb.k10.test_mse <- mean((xgb.k10.test_predictions - xgb.y_test_fold)^2)
  xgb.k10.test_rmse <- sqrt(xgb.k10.test_mse)
  xgb.k10.test_mae <- mean(abs(xgb.k10.test_predictions - xgb.y_test_fold))
  xgb.k10.test_r2 <- 1 - (sum((xgb.y_test_fold - xgb.k10.test_predictions)^2) / sum((xgb.y_test_fold - mean(xgb.y_test_fold))^2))  
  
  # store metrics in the results dataframe
  xgb.k10.results <- rbind(xgb.k10.results, data.frame(
    Fold = i,
    Train_MSE = xgb.k10.train_mse, Test_MSE = xgb.k10.test_mse,
    Train_RMSE = xgb.k10.train_rmse, Test_RMSE = xgb.k10.test_rmse,
    Train_MAE = xgb.k10.train_mae, Test_MAE = xgb.k10.test_mae,
    Train_R2 = xgb.k10.train_r2, Test_R2 = xgb.k10.test_r2
  ))
}

# display the models and their metrics
print(xgb.k10.models)
print(xgb.k10.results)
```
```{r}
# reshape data for plotting
xgb.k10.results_long <- melt(xgb.k10.results, id.vars = 'Fold')

# plot for each metric
p1 <- plot_metric(xgb.k10.results_long, 'MSE')
p2 <- plot_metric(xgb.k10.results_long, 'RMSE')
p3 <- plot_metric(xgb.k10.results_long, 'MAE')
p4 <- plot_metric(xgb.k10.results_long, 'R2')

# arrange the plots in a 2x2 grid
grid.arrange(p1, p2, p3, p4, ncol = 2, nrow = 2)

plot(p1)
plot(p2)
plot(p3)
plot(p4)
```

Decision Trees
```{r}
#### Decision Trees ####

# error in tree: "factor predictors must have at most 32 levels" is thrown

# basically, it becomes computationally expensive to create so many splits in your data, since you are selecting the best split out of all 2^32 (approx) possible splits

# fit a decision tree model on the training data
#tm <- tree(int_rate ~ ., data = train_data)

# make predictions on the training and testing data
#tm.train_predictions <- predict(tm, newdata = train_data)
#tm.test_predictions <- predict(tm, newdata = test_data)

# calculate Mean Squared Error (MSE) for training and testing
#tm.train_mse <- mean((tm.train_predictions - train_data$int_rate)^2)
#tm.test_mse <- mean((tm.test_predictions - test_data$int_rate)^2)

# calculate Root Mean Squared Error (RMSE) for training and testing
#tm.train_rmse <- sqrt(tm.train_mse)
#tm.test_rmse <- sqrt(tm.test_mse)

# calculate Mean Absolute Error (MAE) for training and testing
#tm.train_mae <- mean(abs(tm.train_predictions - train_data$int_rate))
#tm.test_mae <- mean(abs(tm.test_predictions - test_data$int_rate))

# calculate R-squared (R²) for training and testing
#tm.train_r2 <- 1 - (sum((train_data$int_rate - tm.train_predictions)^2) / sum((train_data$int_rate - mean(train_data$int_rate))^2))
#tm.test_r2 <- 1 - (sum((test_data$int_rate - tm.test_predictions)^2) / sum((test_data$int_rate - mean(test_data$int_rate))^2))

# display the metrics
#cat("Training MSE:", tm.train_mse, "\n")
#cat("Testing MSE:", tm.test_mse, "\n")
#cat("Training RMSE:", tm.train_rmse, "\n")
#cat("Testing RMSE:", tm.test_rmse, "\n")
#cat("Training MAE:", tm.train_mae, "\n")
#cat("Testing MAE:", tm.test_mae, "\n")
#cat("Training R-squared (R²):", tm.train_r2, "\n")
#cat("Testing R-squared (R²):", tm.test_r2, "\n")
```

Random Forest
```{r}
#### Random Forest ####

# train a Random Forest model
rf <- ranger(formula = int_rate ~ ., data = train_data, num.trees = 500, verbose=TRUE, importance = "impurity", oob.error = TRUE)

# print the model summary
print("Random Forest Model Summary:")
print(rf)

# make predictions on the training and testing data
rf.train_predictions <- predict(rf, data = train_data)
rf.test_predictions <- predict(rf, data = test_data)

# calculate Mean Squared Error (MSE) for training and testing
rf.train_mse <- mean((rf.train_predictions$predictions - train_data$int_rate)^2)
rf.test_mse <- mean((rf.test_predictions$predictions - test_data$int_rate)^2)

# calculate Root Mean Squared Error (RMSE) for training and testing
rf.train_rmse <- sqrt(rf.train_mse)
rf.test_rmse <- sqrt(rf.test_mse)

# calculate Mean Absolute Error (MAE) for training and testing
rf.train_mae <- mean(abs(rf.train_predictions$predictions - train_data$int_rate))
rf.test_mae <- mean(abs(rf.test_predictions$predictions - test_data$int_rate))

# calculate R-squared (R²) for training and testing
rf.train_r2 <- 1 - (sum((train_data$int_rate - rf.train_predictions$predictions)^2) / sum((train_data$int_rate - mean(train_data$int_rate))^2))
rf.test_r2 <- 1 - (sum((test_data$int_rate - rf.test_predictions$predictions)^2) / sum((test_data$int_rate - mean(test_data$int_rate))^2))

# display the metrics
cat("Training MSE:", rf.train_mse, "\n")
cat("Testing MSE:", rf.test_mse, "\n")
cat("Training RMSE:", rf.train_rmse, "\n")
cat("Testing RMSE:", rf.test_rmse, "\n")
cat("Training MAE:", rf.train_mae, "\n")
cat("Testing MAE:", rf.test_mae, "\n")
cat("Training R-squared (R²):", rf.train_r2, "\n")
cat("Testing R-squared (R²):", rf.test_r2, "\n")
```
Boosting
```{r}
#### Boosting ####

# define the target variable for training and testing
xgb.y_train <- train_data$int_rate
xgb.y_test <- test_data$int_rate

# define the feature matrix for training and testing (exclude the target variable)
xgb.X_train <- train_data[, -which(names(train_data) == 'int_rate')]
xgb.X_test <- test_data[, -which(names(test_data) == 'int_rate')]

# fit a gradient boosting regression model using xgboost
xgb <- xgboost(
  data = as.matrix(xgb.X_train),
  label = xgb.y_train,
  nrounds = 100,
  verbose = 0
)

# make predictions on the training and testing data
xgb.train_predictions <- predict(xgb, newdata = as.matrix(xgb.X_train))
xgb.test_predictions <- predict(xgb, newdata = as.matrix(xgb.X_test))

# calculate Mean Squared Error (MSE) for training and testing
xgb.train_mse <- mean((xgb.train_predictions - xgb.y_train)^2)
xgb.test_mse <- mean((xgb.test_predictions - xgb.y_test)^2)

# calculate Root Mean Squared Error (RMSE) for training and testing
xgb.train_rmse <- sqrt(xgb.train_mse)
xgb.test_rmse <- sqrt(xgb.test_mse)

# calculate Mean Absolute Error (MAE) for training and testing
xgb.train_mae <- mean(abs(xgb.train_predictions - xgb.y_train))
xgb.test_mae <- mean(abs(xgb.test_predictions - xgb.y_test))

# calculate R-squared (R²) for training and testing
xgb.train_r2 <- 1 - (sum((xgb.y_train - xgb.train_predictions)^2) / sum((xgb.y_train - mean(xgb.y_train))^2))
xgb.test_r2 <- 1 - (sum((xgb.y_test - xgb.test_predictions)^2) / sum((xgb.y_test - mean(xgb.y_test))^2))

# display the metrics
cat("Training MSE:", xgb.train_mse, "\n")
cat("Testing MSE:", xgb.test_mse, "\n")
cat("Training RMSE:", xgb.train_rmse, "\n")
cat("Testing RMSE:", xgb.test_rmse, "\n")
cat("Training MAE:", xgb.train_mae, "\n")
cat("Testing MAE:", xgb.test_mae, "\n")
cat("Training R-squared (R²):", xgb.train_r2, "\n")
cat("Testing R-squared (R²):", xgb.test_r2, "\n")
```
Following, a scatter plot of actual vs predicted training values for each model is plot.
This plot helps us visualize how well each model's predictions align with the actual data points.
```{r}
# create a scatter plot function
create_scatter_plot <- function(actual_values, predicted_values, model_name) {
  model_comparison_data <- data.frame(
    Actual = actual_values,
    Predicted = predicted_values
  )
  
  scatter_plot <- ggplot(model_comparison_data, aes(x = Actual, y = Predicted)) +
    geom_point() +
    geom_abline(intercept = 0, slope = 1, linetype = "dashed", color = "red") +  # add a diagonal reference line
    labs(x = "Actual Training Values", y = "Predicted Training Values", title = model_name) +
    theme_minimal() +
    ylim(-50, 50)
  
  return(scatter_plot)
}

# create scatter plots for each model
lm_scatter_plot <- create_scatter_plot(
  actual_values = train_data$int_rate,
  predicted_values = lm.train_predictions,
  model_name = "Linear Regression"
)

rf_scatter_plot <- create_scatter_plot(
  actual_values = train_data$int_rate,
  predicted_values = rf.train_predictions$predictions,
  model_name = "Random Forest"
)

xgb_scatter_plot <- create_scatter_plot(
  actual_values = xgb.y_train,
  predicted_values = xgb.train_predictions,
  model_name = "XGBoost"
)

# display the scatter plots separately
print(lm_scatter_plot)
print(rf_scatter_plot)
print(xgb_scatter_plot)
```
Following, a scatter plot of actual vs predicted testing values for each model is plot.
This plot helps us visualize how well each model's predictions align with the actual data points.
```{r}
# create a scatter plot function
create_scatter_plot <- function(actual_values, predicted_values, model_name) {
  model_comparison_data <- data.frame(
    Actual = actual_values,
    Predicted = predicted_values
  )
  
  scatter_plot <- ggplot(model_comparison_data, aes(x = Actual, y = Predicted)) +
    geom_point() +
    geom_abline(intercept = 0, slope = 1, linetype = "dashed", color = "red") +  # add a diagonal reference line
    labs(x = "Actual Testing Values", y = "Predicted Testing Values", title = model_name) +
    theme_minimal() +
    ylim(-50, 50) +
    xlim(0, 40)
  
  return(scatter_plot)
}

# create scatter plots for each model
lm_scatter_plot <- create_scatter_plot(
  actual_values = test_data$int_rate,
  predicted_values = lm.test_predictions,
  model_name = "Linear Regression"
)

rf_scatter_plot <- create_scatter_plot(
  actual_values = test_data$int_rate,
  predicted_values = rf.test_predictions$predictions,
  model_name = "Random Forest"
)

xgb_scatter_plot <- create_scatter_plot(
  actual_values = xgb.y_test,
  predicted_values = xgb.test_predictions,
  model_name = "XGBoost"
)

# display the scatter plots separately
print(lm_scatter_plot)
print(rf_scatter_plot)
print(xgb_scatter_plot)
```

Residual plots can help identify patterns in prediction errors and assess whether the assumptions of linear regression (if applicable) are met.
```{r}
# create a residual plot function
create_residual_plot <- function(actual_values, predicted_values, model_name) {
  residuals <- actual_values - predicted_values
  residual_data <- data.frame(
    Predicted = predicted_values,
    Residuals = residuals
  )
  
  residual_plot <- ggplot(residual_data, aes(x = Predicted, y = Residuals)) +
    geom_point() +
    geom_hline(yintercept = 0, linetype = "dashed", color = "red") +  # Red horizontal reference line
    labs(x = "Predicted Values", y = "Residuals", title = paste("Residual Plot -", model_name)) +
    theme_minimal() +
    ylim(-30, 30) +
    xlim(0, 40)
  
  return(residual_plot)
}

# create residual plots for each model
lm_residual_plot <- create_residual_plot(
  actual_values = train_data$int_rate,
  predicted_values = lm.train_predictions,
  model_name = "Linear Regression"
)

rf_residual_plot <- create_residual_plot(
  actual_values = train_data$int_rate,
  predicted_values = rf.train_predictions$predictions,
  model_name = "Random Forest"
)

xgb_residual_plot <- create_residual_plot(
  actual_values = xgb.y_train,
  predicted_values = xgb.train_predictions,
  model_name = "XGBoost"
)

# display the residual plots separately
print(lm_residual_plot)
print(rf_residual_plot)
print(xgb_residual_plot)
```

```{r}
# create a density plot function for residuals
create_residual_density_plot <- function(actual_values, predicted_values, model_name) {
  residuals <- actual_values - predicted_values
  residual_data <- data.frame(Residuals = residuals)
  
  density_plot <- ggplot(residual_data, aes(x = Residuals)) +
    geom_density(fill = "skyblue", color = "black", alpha = 0.7) +
    labs(x = "Residuals", y = "Density", title = paste("Residual Density Plot -", model_name)) +
    theme_minimal() +
    xlim(-30,30) + 
    ylim(0, 0.35)
    
  
  return(density_plot)
}

# create density plots for residuals for each model
lm_residual_density_plot <- create_residual_density_plot(
  actual_values = train_data$int_rate,
  predicted_values = lm.train_predictions,
  model_name = "Linear Regression"
)

rf_residual_density_plot <- create_residual_density_plot(
  actual_values = train_data$int_rate,
  predicted_values = rf.train_predictions$predictions,
  model_name = "Random Forest"
)

xgb_residual_density_plot <- create_residual_density_plot(
  actual_values = xgb.y_train,
  predicted_values = xgb.train_predictions,
  model_name = "XGBoost"
)

# display the density plots separately
print(lm_residual_density_plot)
print(rf_residual_density_plot)
print(xgb_residual_density_plot)
```

This visualization can help you compare the distribution of prediction errors across models through histograms.

```{r}
# create a histogram plot function for residuals with a red density curve
create_residual_histogram_plot <- function(actual_values, predicted_values, model_name) {
  residuals <- actual_values - predicted_values
  residual_data <- data.frame(Residuals = residuals)
  
  histogram_plot <- ggplot(residual_data, aes(x = Residuals)) +
    geom_histogram(aes(y = after_stat(density)), bins = 30, fill = "skyblue", color = "black", alpha = 0.7) +  # use density on the y-axis for the histogram
    geom_density(color = "red", linewidth = 1.5) +  # add the density plot in red
    labs(x = "Residuals", y = "Density", title = paste("Residual Histogram Plot with Density Curve -", model_name)) +
    theme_minimal() +
    xlim(-20,20) + 
    ylim(0, 0.3)
  
  return(histogram_plot)
}

# create histogram plots for residuals for each model
lm_residual_histogram_plot <- create_residual_histogram_plot(
  actual_values = train_data$int_rate,
  predicted_values = lm.train_predictions,
  model_name = "Linear Regression"
)

rf_residual_histogram_plot <- create_residual_histogram_plot(
  actual_values = train_data$int_rate,
  predicted_values = rf.train_predictions$predictions,
  model_name = "Random Forest"
)

xgb_residual_histogram_plot <- create_residual_histogram_plot(
  actual_values = xgb.y_train,
  predicted_values = xgb.train_predictions,
  model_name = "XGBoost"
)

# display the histogram plots separately
print(lm_residual_histogram_plot)
print(rf_residual_histogram_plot)
print(xgb_residual_histogram_plot)
```

For each model a bar chart that displays the R-squared (coefficient of determination) values is created.
R-squared measures the proportion of variance in the target variable explained by the model. Higher R-squared values indicate better model fit.
```{r}
# create a data frame with R-squared values for each model
model_names <- c("Linear Regression", "Random Forest", "XGBoost")
r_squared_values_train <- c(
  lm.train_r2,
  rf.train_r2,
  xgb.train_r2
)
r_squared_values_test <- c(
  lm.test_r2,
  rf.test_r2,
  xgb.test_r2
)

r_squared_data_train <- data.frame(Model = factor(model_names),
                              R_squared = r_squared_values_train)
r_squared_data_test <- data.frame(Model = factor(model_names),
                              R_squared = r_squared_values_test)

# create the R-squared comparison bar chart
r_squared_bar_chart_train <- ggplot(r_squared_data_train, aes(x = Model, y = R_squared, fill = Model)) +
  geom_bar(stat = "identity") +
  labs(x = "Model", y = "R-squared (R²)", title = "R-squared Comparison Training") +
  theme_minimal() +
  theme(axis.text.x = element_text(angle = 45, hjust = 1)) + 
  ylim(0,1)
r_squared_bar_chart_test <- ggplot(r_squared_data_test, aes(x = Model, y = R_squared, fill = Model)) +
  geom_bar(stat = "identity") +
  labs(x = "Model", y = "R-squared (R²)", title = "R-squared Comparison Testing") +
  theme_minimal() +
  theme(axis.text.x = element_text(angle = 45, hjust = 1)) + 
  ylim(0,1)

# display the R-squared comparison bar chart
print(r_squared_bar_chart_train)
print(r_squared_bar_chart_test)
```
A bar chart that compares the MAE or RMSE values, is generated for each model.
These metrics quantify the average prediction errors of each model, and lower values are preferred.
```{r}
# create a data frame with MAE and RMSE values for each model
model_names <- c("Linear Regression", "Random Forest", "XGBoost","Linear Regression", "Random Forest", "XGBoost")
error_values_train <- c(
  lm.train_mae,
  rf.train_mae,
  xgb.train_mae,
  lm.train_rmse,
  rf.train_rmse,
  xgb.train_rmse
)
error_values_test <- c(
  lm.test_mae,
  rf.test_mae,
  xgb.test_mae,
  lm.test_rmse,
  rf.test_rmse,
  xgb.test_rmse
)
error_type <- c(
  "MAE", "MAE", "MAE","RMSE","RMSE","RMSE"
)
model_errors_train <- data.frame(Model = factor(model_names, levels = c("Linear Regression", "Random Forest", "XGBoost")),
                Error = error_values_train, Type = error_type)
model_errors_test <- data.frame(Model = factor(model_names, levels = c("Linear Regression", "Random Forest", "XGBoost")),
                Error = error_values_test, Type = error_type)
# create the MAE or RMSE comparison bar chart
error_bar_chart_train <- ggplot(model_errors_train, aes(x = Model, y = Error, fill = Type)) +
  geom_bar(stat = "identity", position = "dodge") +
  labs(x = "Model", y = "Error Value", title = "Training MAE and RMSE Comparison") +
  theme_minimal() +
  theme(axis.text.x = element_text(angle = 45, hjust = 1)) + 
  ylim(0, 4)

error_bar_chart_test <- ggplot(model_errors_test, aes(x = Model, y = Error, fill = Type)) +
  geom_bar(stat = "identity", position = "dodge") +
  labs(x = "Model", y = "Error Value", title = "Testing MAE and RMSE Comparison") +
  theme_minimal() +
  theme(axis.text.x = element_text(angle = 45, hjust = 1)) + 
  ylim(0, 4)

# display the MAE and RMSE comparison bar chart
print(error_bar_chart_train)
print(error_bar_chart_test)
```


```{r}
#### Random Forest Feature Importance Plot ####
v1 <- vip(rf, title = "Ranger", num_features = 20) 
plot(v1)
```
Feature Selection from the variable importance's analysis:

```{r}
imp.variables <- lc_data[, v1$data$Variable]
imp.variables$int_rate <- lc_data$int_rate
imp.train_indices <- createDataPartition(imp.variables$int_rate, p = 0.8, list = FALSE)

# create training and testing datasets
imp.train_data <- imp.variables[imp.train_indices, ]
imp.test_data <- imp.variables[-imp.train_indices, ]
```

```{r}
#### Linear Regression with only importance variables ####

imp.lm.fit <- lm(int_rate ~ ., data = imp.train_data)

# make predictions on the training and testing data
imp.lm.train_predictions <- predict(imp.lm.fit, newdata = imp.train_data)
imp.lm.test_predictions <- predict(imp.lm.fit, newdata = imp.test_data)

# calculate Mean Squared Error (MSE) for training and testing
imp.lm.train_mse <- mean((imp.lm.train_predictions - imp.train_data$int_rate)^2)
imp.lm.test_mse <- mean((imp.lm.test_predictions - imp.test_data$int_rate)^2)

# calculate Root Mean Squared Error (RMSE) for training and testing
imp.lm.train_rmse <- sqrt(imp.lm.train_mse)
imp.lm.test_rmse <- sqrt(imp.lm.test_mse)

# calculate Mean Absolute Error (MAE) for training and testing
imp.lm.train_mae <- mean(abs(imp.lm.train_predictions - imp.train_data$int_rate))
imp.lm.test_mae <- mean(abs(imp.lm.test_predictions - imp.test_data$int_rate))

# calculate R-squared (R²) for training and testing
imp.lm.train_r2 <- 1 - (sum((imp.train_data$int_rate - imp.lm.train_predictions)^2) / sum((imp.train_data$int_rate - mean(imp.train_data$int_rate))^2))
imp.lm.test_r2 <- 1 - (sum((imp.test_data$int_rate - imp.lm.test_predictions)^2) / sum((imp.test_data$int_rate - mean(imp.test_data$int_rate))^2))

# display the metrics
cat("Training MSE:", imp.lm.train_mse, "\n")
cat("Testing MSE:", imp.lm.test_mse, "\n")
cat("Training RMSE:", imp.lm.train_rmse, "\n")
cat("Testing RMSE:", imp.lm.test_rmse, "\n")
cat("Training MAE:", imp.lm.train_mae, "\n")
cat("Testing MAE:", imp.lm.test_mae, "\n")
cat("Training R-squared (R²):", imp.lm.train_r2, "\n")
cat("Testing R-squared (R²):", imp.lm.test_r2, "\n")
```

```{r}
#### Random Forest with only importance variables ####

# train a Random Forest model
imp.rf <- ranger(formula = int_rate ~ ., data = imp.train_data, num.trees = 500, verbose=TRUE, importance = "impurity", oob.error = TRUE)

# print the model summary
print("Random Forest Model Summary:")
print(imp.rf)

# make predictions on the training and testing data
imp.rf.train_predictions <- predict(imp.rf, data = imp.train_data)
imp.rf.test_predictions <- predict(imp.rf, data = imp.test_data)

# calculate Mean Squared Error (MSE) for training and testing
imp.rf.train_mse <- mean((imp.rf.train_predictions$predictions - imp.train_data$int_rate)^2)
imp.rf.test_mse <- mean((imp.rf.test_predictions$predictions - imp.test_data$int_rate)^2)

# calculate Root Mean Squared Error (RMSE) for training and testing
imp.rf.train_rmse <- sqrt(imp.rf.train_mse)
imp.rf.test_rmse <- sqrt(imp.rf.test_mse)

# calculate Mean Absolute Error (MAE) for training and testing
imp.rf.train_mae <- mean(abs(imp.rf.train_predictions$predictions - imp.train_data$int_rate))
imp.rf.test_mae <- mean(abs(imp.rf.test_predictions$predictions - imp.test_data$int_rate))

# calculate R-squared (R²) for training and testing
imp.rf.train_r2 <- 1 - (sum((imp.train_data$int_rate - imp.rf.train_predictions$predictions)^2) / sum((imp.train_data$int_rate - mean(imp.train_data$int_rate))^2))
imp.rf.test_r2 <- 1 - (sum((test_data$int_rate - rf.test_predictions$predictions)^2) / sum((imp.test_data$int_rate - mean(imp.test_data$int_rate))^2))

# display the metrics
cat("Training MSE:", imp.rf.train_mse, "\n")
cat("Testing MSE:", imp.rf.test_mse, "\n")
cat("Training RMSE:", imp.rf.train_rmse, "\n")
cat("Testing RMSE:", imp.rf.test_rmse, "\n")
cat("Training MAE:", imp.rf.train_mae, "\n")
cat("Testing MAE:", imp.rf.test_mae, "\n")
cat("Training R-squared (R²):", imp.rf.train_r2, "\n")
cat("Testing R-squared (R²):", imp.rf.test_r2, "\n")
```

```{r}
#### Boosting with only importance variables ####

# define the target variable for training and testing
imp.xgb.y_train <- imp.train_data$int_rate
imp.xgb.y_test <- imp.test_data$int_rate

# define the feature matrix for training and testing (exclude the target variable)
imp.xgb.X_train <- imp.train_data[, -which(names(imp.train_data) == 'int_rate')]
imp.xgb.X_test <- imp.test_data[, -which(names(imp.test_data) == 'int_rate')]

# fit a gradient boosting regression model using xgboost
imp.xgb <- xgboost(
  data = as.matrix(imp.xgb.X_train),
  label = imp.xgb.y_train,
  nrounds = 100,
  verbose = 0
)

# make predictions on the training and testing data
imp.xgb.train_predictions <- predict(imp.xgb, newdata = as.matrix(imp.xgb.X_train))
imp.xgb.test_predictions <- predict(imp.xgb, newdata = as.matrix(imp.xgb.X_test))

# calculate Mean Squared Error (MSE) for training and testing
imp.xgb.train_mse <- mean((imp.xgb.train_predictions - imp.xgb.y_train)^2)
imp.xgb.test_mse <- mean((imp.xgb.test_predictions - imp.xgb.y_test)^2)

# calculate Root Mean Squared Error (RMSE) for training and testing
imp.xgb.train_rmse <- sqrt(imp.xgb.train_mse)
imp.xgb.test_rmse <- sqrt(imp.xgb.test_mse)

# calculate Mean Absolute Error (MAE) for training and testing
imp.xgb.train_mae <- mean(abs(imp.xgb.train_predictions - imp.xgb.y_train))
imp.xgb.test_mae <- mean(abs(imp.xgb.test_predictions - imp.xgb.y_test))

# calculate R-squared (R²) for training and testing
imp.xgb.train_r2 <- 1 - (sum((imp.xgb.y_train - imp.xgb.train_predictions)^2) / sum((imp.xgb.y_train - mean(imp.xgb.y_train))^2))
imp.xgb.test_r2 <- 1 - (sum((imp.xgb.y_test - imp.xgb.test_predictions)^2) / sum((imp.xgb.y_test - mean(imp.xgb.y_test))^2))

# display the metrics
cat("Training MSE:", imp.xgb.train_mse, "\n")
cat("Testing MSE:", imp.xgb.test_mse, "\n")
cat("Training RMSE:", imp.xgb.train_rmse, "\n")
cat("Testing RMSE:", imp.xgb.test_rmse, "\n")
cat("Training MAE:", imp.xgb.train_mae, "\n")
cat("Testing MAE:", imp.xgb.test_mae, "\n")
cat("Training R-squared (R²):", imp.xgb.train_r2, "\n")
cat("Testing R-squared (R²):", imp.xgb.test_r2, "\n")
```

The dataset was filtered by the 20 variables with the most importance (from the rf results). As we can see above, the errors of each model are more or less the errors with the double variables we had before, so filtering by these 20 "important variables" does not seem making sense...

Hyperparameter Tuning for XGBoosting:

```{r}
# define the number of cores
numCores <- detectCores() - 1

# register doParallel as the backend for parallel execution
registerDoParallel(cores=numCores)

# define the control using a cross-validation approach
train_control <- trainControl(method = "cv", number = 5, verboseIter = TRUE)

# define the grid of hyperparameters to search over
xgb.grid <- expand.grid(
  nrounds = c(100, 200, 300),
  eta = c(0.01, 0.05, 0.1),
  max_depth = c(3, 6, 9),
  gamma = c(0, 0.1, 0.2),
  colsample_bytree = c(0.5, 0.8, 1),
  min_child_weight = c(1, 5, 10),
  subsample = c(0.5, 0.75, 1)
)

# train the model
xgb.tuned <- train(
  x = train_data, y = xgb.y_train,
  method = "xgbTree",
  trControl = train_control,
  tuneGrid = xgb.grid
)

# view the best tuning parameters
print(xgb.tuned$bestTune)

# stop the parallel backend
stopImplicitCluster()
```